Broadcasting of Entanglement and Consequences on Quantum Information Processing

Transcription

1 Broadcasting of Entanglement and Consequences on Quantum Information Processing Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science by Research in Computational Natural Sciences by Jaya International Institute of Information Technology, Hyderabad (Deemed to be University) Hyderabad , INDIA Dec 017

2 Copyright c Jaya, 017 All Rights Reserved

3 International Institute of Information Technology Hyderabad, India CERTIFICATE It is certified that the work contained in this thesis, titled Broadcasting of Entanglement and Consequences on Quantum Information Processing by Jaya( ), has been carried out under my supervision and is not submitted elsewhere for a degree. Date Adviser: Dr. Indranil Chakrabarty Co-adviser: Dr. Prabhakar Bhimalapuram

4 Dedicated to my parents

5 Acknowledgments It is a great pleasure to express my heartfelt gratitude to my advisor Dr. Indranil Chakrabarty for initiating the idea for my research work. His sustained advice and guidance till the completion of this thesis were really helpful. His patience and positive attitude were the key factors which kept me going all through this journey. I would also like to express my sincere appreciation to my co-advisor Dr. Prabhakar Bhimalapuram whose encouragement and suggestions catalysed my learning process. I am really thankful to him for being very understanding and counseling me time to time both academically and personally. I would also take this opportunity to thank the Center for Computational Natural Sciences and Bioinformatics, International Institute of Information Technology, Hyderabad for assisting me in carrying out this research work. Many thanks to all the CCNSB faculty members for building the background of my research through the classroom courses. I would like to give my special thanks to Dr. Harjinder Singh, Dr. Abhijit Mitra and Dr. Deva Priyakumar for being considerate and supportive to me always. The discussions with my fellow group members was the great source of learning and getting insights into the field of quantum information and quantum computations, for which I would like to mention Sourav, Palash, Maharshi, Anubhav, Manish and others. Special thanks to Sourav for conducting prolonged discussion sessions and clearing my doubts, which really grew my interests even more in this work. Apart from the research work, the people who really helped me out in one form or the other throughout this journey are Pranneetha, Supriya, Indrajit, Namra, Arpita, Avni, Prateek and many more. And last but not the least, the constant unconditional support of my parents is the foundation of everything I have. This work would never have been possible without their support, blessings and encouragement. v

6 Abstract Information theory plays a vital role in today s computational world. According to the past research, the capabilities of the quantum information processing exceeds that of the classical methods in most of the cases. However, there are few phenomena where quantum mechanics puts limitations such as cloning information. In classical computers, we can copy the information from one place to another without any alteration. But, we cannot achieve the similar results following the quantum rules. This limitations of the quantum mechanics comes from no cloning theorem which says that it is impossible to copy an arbitrary pure state (information) perfectly. But the possibility of an imperfect cloning is never denied. Broadly, two types of quantum cloning machines have been discussed in past: approximate cloning machine, which gives imperfect copies and probabilistic cloning machines, which gives a perfect copy but only with some non-unit probability. The quality of the state cloned is measured in terms of fidelity of cloning. The approximate cloning machines, whose fidelity depends on the input state, are called state dependent cloning machines. However, the state independent cloning machine copies any state with the same fidelity. In this work, we talk about the state independent approximate cloning machines. We extend the already existing 1 to N local cloning machine and give the cloning transformations to apply it non-locally. Along with the cloning of a state, one more phenomena is of interest here, that is, cloning of entanglement. Quantum entanglement is one of the peculiarities of quantum mechanics, which makes the phenomena such as quantum teleportation and super-dense coding possible. When we refer broadcasting of an entangled state, we mean creating more pairs of less entangled state from a given entangled state. One way of doing this is by applying local cloning transformations on each qubit of the given entangled state. This can also be done by applying global cloning operations on the entangled state itself. It is known that the broadcasting of entanglement into more than two entangled pairs is not possible using only local operations. Also, Inseparability is optimally broadcast when symmetric cloners are applied. This is mostly required when we perform distributed information processing tasks. It is natural to expect in a bi-partite situation that these newly created entangled states are less suitable in tasks like teleportation and super-dense coding than the parent state. However, the dependence of the usability of such states in information processing tasks on our ability of broadcasting is not well known. In this work we obtain several complementary relations manifesting the interdependence of their information processing capabilities and fidelity of broadcasting. We extend our investigation in a situation where vi

7 vii instead of using 1- cloning, we have used 1-N cloning transformations both locally and non-locally. Eventually we find out the fidelity of broadcasting and the change in the information processing capacities for different values of N(3,4,5) and investigate how these complementary relations behave with the increase in number of copies(n).

8 Contents Chapter Page 1 Introduction History Mathematical foundations in Quantum Mechanics Probability Vectors, Basis and Vector Space Dirac notation Outer product Quantum Bit Matrix and linear operators Eigenvectors and eigenvalues Inner products Tensor products Operators Bloch sphere State vector The density matrix Trace The reduced density operator and partial trace Pauli Matrices Postulates of Quantum Mechanics First postulate - Quantum State Second Postulate - Hilbert space Third Postulate - Operators Linear Operator : Hermitian Operator : Fourth postulate - Evolution Fifth postulate - Expectation value Quantum Information and Entropy Shannon Entropy Von Neumann Entropy Quantum Correlation Entanglement and Separability condition Peres-Horodecki Criterion Entangled states viii

9 CONTENTS ix 1.6 Entanglement assisted information processing tasks Quantum Teleportation Superdense Coding Quantum cloning No Cloning Theorem No Broadcast Theorem Complementarity Cloning and Broadcasting No cloning theorem Probabilistic cloning Approximate cloning State Dependent cloner Wootters-Zurek cloner Fidelity of cloning State Independent cloner Buzek-Hillery cloner Optimality of cloning machine To M copies - cloning machine Gisin-Massar cloning machine Broadcasting No broadcast theorem Broadcasting of Entanglement Broadcasting via local cloning To M local cloner Broadcasting qubits most general state Impossibility of cloning more than two copies Broadcasting via non-local cloning Buzek-Hillery 1 non-local cloner to M non-local cloner Derivation of Buzek-Hillery non-local cloner qubits Most General State - broadcasting two copies Increasing the number of copies Conclusion Complementarity of Information Processing Tasks with Broadcasting Fidelity Complementarity in Quantum mechanics Complementarity in Quantum information Complementarity between broadcasting fidelity and Information Processing Tasks Complementarity for Two qubits generalized mixed states Complementarity in Broadcasting multiple copies Conclusions Bibliography

10 List of Figures Figure Page 1.1 Sequence of Stern Gerlach Experiment for demonstrating quantum behavior of electrons Experiment setup for demonstrating randomness of photon Polar Coordinates Representation of a complex number Polar Coordinates Representation of a complex number in 3D Pictorial representation of local cloning Pictorial representation of non-local cloning a) Plotting the sum DC + F B with T r[ρ ]. b) Plotting the sum T F + F B with T r[ρ ]. In each of the cases the cloning operation applied is local cloning a) Plotting the sum DC + F B with T r[ρ ]. b) Plotting the sum T F + F B with T r[ρ ]. In each of the cases the cloning operation applied is non local cloning Plotting F B with α. The cloning operation applied is local cloning a) Plotting F B with T F. b) Plotting F B with DC In each of the cases the cloning operation applied is local cloning a) Plotting F B + T F with α. b) Plotting F B + DC with α In each of the cases the cloning operation applied is local cloning Plotting F B with α. The cloning operation applied is non-local cloning a) Plotting F B with T F. b) Plotting F B with DC In each of the cases the cloning operation applied is non-local cloning a) Plotting F B + T F with α. b) Plotting F B + DC with α In each of the cases the cloning operation applied is non-local cloning Plotting F B vs p vs α. The cloning operation applied is local cloning a) Plotting F B with T F. b) Plotting F B with DC In each of the cases the cloning operation applied is local cloning a) Plotting ( T F + F B) and b) ( DC + F B) against the parameters α and p of the Werner like state. These are obtained using local cloner Plotting F B vs p vs α. The cloning operation applied is non-local cloning a) Plotting F B with T F. b) Plotting F B with DC In each of the cases the cloning operation applied is non-local cloning a) Plotting ( T F + F B) and b) ( DC + F B) against the parameters α and p of the Werner like state. These are obtained using non-local cloner x

11 List of Tables Table Page 3.1 Broadcasting ranges obtained using local cloners for different values of α Broadcasting ranges obtained with local cloners for different values of p The broadcasting ranges obtained with a nonlocal cloner for different values of the input state parameter (α ) The broadcasting ranges obtained with a nonlocal cloner for different values of the classical mixing parameter (p) Broadcasting ranges along with the maximum values of the sums ( T F +F B)max and ( DC +F B)max obtained with local cloners for different valid values of input state parameters c 1 and c Broadcasting ranges along with the maximum values of the sums ( T F +F B)max and ( DC +F B)max obtained with non local cloners for different valid values of input state parameters c 1 and c [Werner Like States] The complementarity bound for teleportation fidelity and broadcasting fidelity obtained with a 1 M non local cloner for different values of the input state parameter(α ) where M =, 3, 4, [Werner Like States] The complementarity bound for teleportation fidelity and broadcasting fidelity obtained with a 1 M non local cloner for different values of the input state parameter(p) where M =, 3, 4, [Werner Like States] The complementarity bound for super dense coding capacity and broadcasting fidelity obtained with a 1 M non local cloner for different values of the input state parameter(α ) where M =, 3, 4, [Werner Like States] The complementarity bound for super dense coding capacity and broadcasting fidelity obtained with a 1 M local cloner for different values of the input state parameter(p) where M =, 3, 4, [Bell Diagonal States] The complementarity bound for teleportation fidelity and broadcasting fidelity obtained with a 1 M local cloner for different values of the input state parameter(c, c 3 ) where M =, 3, 4, [Bell Diagonal States] The complementarity bound for super dense coding capacity and broadcasting fidelity obtained with a 1 M non local cloner for different values of the input state parameter(c, c 3 ) where M =, 3, 4, xi

12 Chapter 1 Introduction Contents 1.1 History Mathematical foundations in Quantum Mechanics Probability Vectors, Basis and Vector Space Dirac notation Outer product Quantum Bit Matrix and linear operators Eigenvectors and eigenvalues Inner products Tensor products Operators Bloch sphere State vector The density matrix Trace The reduced density operator and partial trace Pauli Matrices Postulates of Quantum Mechanics First postulate - Quantum State

13 1.3. Second Postulate - Hilbert space Third Postulate - Operators Fourth postulate - Evolution Fifth postulate - Expectation value Quantum Information and Entropy Shannon Entropy Von Neumann Entropy Quantum Correlation Entanglement and Separability condition Entangled states Entanglement assisted information processing tasks Quantum Teleportation Superdense Coding Quantum cloning No Cloning Theorem No Broadcast Theorem Complementarity Mathematics is the language in which the gods speak to people. -Plato

14 1.1 History In classical mechanics, there is a set of rules defined according to which a system evolves and if we measure some property of the system like its position or momentum, we will get an expected value and the system will remain in the expected state. This perception was generalized to any system till some weird behaviour of small sized particles such as electrons and photons were observed. It all started with an experiment performed by the German physicists Otto Stern and Walter Gerlach performed in 19, in which they discovered the unexpected behaviour of electrons [50]. Later, they found that the quantum system no longer remain in their original state after a measurement is performed on it through the following experiment. Sequential Stern-Gerlach Experiment The Stern-Gerlach experiment was performed with a beam of silver electrons travelling through an inhomogeneous magnetic field. The aim of the experiment was to measure the spin of the particle. Because of the rotating nature of electrons as in Bohr s model of atom, we could expect the atoms to get deflected in all possible directions. But, to the surprise the atoms were deflected in only two directions, which proved that the direction of the angular momentum of electrons is quantized. Further, an experiment with a sequence of Stern-Gerlach(SG) apparatus showed that the measurement of spin in one direction changes the same in other direction [86]. Figure 1.1 Sequence of Stern Gerlach Experiment for demonstrating quantum behavior of electrons In the figure 1.1 the rectangular boxes are the pairs of magnetic poles. The straight lines show the flow of atoms. The first part of this experimental set-up is same as of the original SG apparatus to measure the spin along z-axis. In the second step, electrons having negative spin from the first step were blocked and the remaining electrons were passed to the magnetic field in perpendicular direction to the earlier 3

15 i.e. x-axis. Now again, we see two beams coming out, one having magnetic moment in positive x-axis and the other in the negative x-axis. Again when we pass through the third splitter which is along z-axis, we get the beam split into two parts, both positive and negative z-spins. This phenomena was explained as, any kind of measurement on an object changes the state of the object. When we measured the spin along x direction, the z component got tampered. That s why even after filtering out the negative z component we are able to retrieve it again. This tampering can happen randomly i.e. measurement along any axis can affect any other. Why quantum computation? By understanding and harnessing the laws of quantum mechanics we can solve so many computational problems in existing classical computers. For example, the simulations of quantum mechanical processes in physics, chemistry or Biology need a large computer memory. In classical computers information is stored as a bit which can have only two values 0 or 1. On the other hand, a qubit can have a range of values between 0 and 1 states on bloch sphere. So, the memory issue is one of the problems which can be solved using quantum computers. Factoring large number is considered as impossible in classical computers till now. However, given a fully functional quantum computer, it is possible to factor a very large number, as claimed by Peter Shor, a mathematician from the Massachusetts Institute of Technology (MIT)[93]. Pecularity of quantum mechanics Two phenomena in quantum mechanics are extremely interesting and useful : Superposition and Entanglement [65] [66]. We are not able to experience these behaviours because they happen at the atomic level involving electrons, photons etc. In the simplest term the superposition is the ability of a quantum system to be in two different states at the same time. For example, a bit having 0 and 1 value simultaneously and after measurement we will get either of the two values. Entanglement is the property of a quantum system in which the two particles are correlated even being at a great distance. Einstein described it as Spooky action at a distance [5]. Along with providing the vast memory, a quantum computer can do a large number of calculation simultaneously because of entanglement and superposition. Now, we believe that quantum mechanics has lots of benefits over classical mechanics. But it has also got some limitations like a perfect cloning is never possible in quantum world. However, it is possible to have imperfect cloning with some non-unit fidelity, which basically denotes the distance between two states. The other peculiarities of quantum mechanics come from Indeterminism, Interference and Uncertainty. Indeterminism is about not being able to determine the state of a system by a set of physical quantities unlike for classical systems, where position and momentum can exactly tell about the state of a particle. Interference phenomena came into picture after the famous double-slit experiment, which gave evidences for the wave-particle duality [4] present in electrons. The Uncertainty [59] principle says that position and momentum of a particle cannot be 4

16 measured simultaneously with arbitrarily high precision even with perfect instruments and techniques. 1. Mathematical foundations in Quantum Mechanics The great insights into quantum information and quantum computations can be achieved through studying mathematical structure of quantum mechanics. A physical system can be described appropriately using state, observable and dynamics. Von Neumann has contributed significantly in preparing the mathematical foundation of quantum mechanics. He coined a term Hilbert Space which generalizes the notion of Euclidean space with infinite number of dimensions. A Hilbert Space includes vector space and inner product space. Measurement is also one of important phenomena in any quantum system. We don t know the exact state of the system unless we perform a series of calculations. That is why predefined mathematical formalism plays a very important role in making the process to be carried out with a pen and a paper. There are so many complex phenomena in quantum mechanics like superposition, entanglement, super dense coding etc. To perform complex operations and to represent all these on paper, we need a fair way which can be used throughout the scientific community. When quantum mechanics was initially developed, the mathematical calculations involving matrices and matrix operations were not the part of it. After Dirac introduced his own approach to quantum mechanics, the world started finding it easier to follow. He gave the corresponding quantum mechanical notations for the standard matrix algebra [43] Probability Classical mechanics is deterministic whereas quantum mechanics is probabilistic. In classical mechanics, probability is often linked with the randomness present in any system. When we toss a fair coin, there is half probability to get either head or tail every time we toss it. It seems to be impossible to find out what comes next but it is not. Here we are ignoring the fact that this phenomenon of tossing a coin is just a result of a list of forces acting on it. Given that we maintain the magnitude and the direction of all the forces being applied on the coin and also the environment condition is same every time we toss it, we will get the same output. So, for this particular example we can say that the uncertainty arises not because of randomness in the system but because of the human inability to maintain the integrity of all the forces. The actual example of the uncertainty arising due to randomness can be seen in Quantum mechanics, where a state of a particle can have a range of values(positions and momenta) and we can t know which value it is holding unless we measure it. Also, every time we measure, it might give different value because after the measurement the state collapses to an eigenstate of the operator applied on it. We see, randomness is the fundamental property of a quantum mechanical system. Here, the probability 5

17 theories play an important role to understand the quantum system. An appropriate experiment depicting this inherent characteristic of matter is described in the book Quantum Theory: Concepts and Methods by Asher Peres [8]. The experiment set-up is as follows: Figure 1. Experiment setup for demonstrating randomness of photon There is a light source S, a polarizer P, a pinhole H, a calcite crystal C and a detector D. The light from S passes through P and therefore the polarized light goes through crystal C which is a birefringent crystal. Finally, light falls on the screen forming two spots of light usually of different brightness. As the polarizer is rotated with respect to the crystal by an angle α, then the intensities of spots vary as cos α and sin α. Because the light consists of photons and each photon is indivisible, we can t get split beams having energies hν cos α and hν sin α corresponding to each spot. Rather, we get fewer photons with full energy hν. If the arrivals of photons on detector are recorded by printing + and depending on whether upper or lower detector was triggered respectively. Then, we get a random sequence of + and. If N + and N are the total numbers of + and respectively, then N + N + +N and N N + +N are the probabilities tending to cos α and sin α respectively as N + and N become large. Also, there is no explanation that we can predict the next printout. It is totally probabilistic. So, the conclusion is that as long as we accept that the polarized light consists of photons and the photon is indivisible, randomness becomes fundamental. 1.. Vectors, Basis and Vector Space Generally speaking, vectors are the mathematical objects which have both the magnitude and the direction. The most primitive type of vector is used to locate a position in the space. We can take Linear Combinations of a number of vectors to get a new vector. Two vectors are said to be linearly independent if no multiples of these vectors combine to give a zero vector. A Basis is the set of vectors which span over the entire space i.e. any vector can be written as the linear combination of these vectors. Mathematically, a vector is represented as a û1 + b ˆ u, where a and b are real constants and ˆ u1 and ˆ u are the basis vectors in dimensional space. The dimension of the space can go even higher than 3 depending on the type of vectors. Also, the constants 6

18 like a and b can be complex numbers. It is easy to visualize the position vector as a real vector. So, it could be generalized to other abstract quantities, which has nothing to do with the position vector but help us understand the analogous properties. In Quantum Mechanics the vectors are used to represent a state. However, the mathematical behavior of those remain the same. The vector space is the collection of all possible vectors Dirac notation Dirac introduced the entities called bra and ket which are nothing but row vector and column vector respectively. The linear operators are represented as a square matrix. The elements of these vectors and matrices are generally complex numbers. A bra is denoted as A and a ket is denoted as A. For a system of 3 dimensions, bra vector A, ket vector B and an operator α can be written in matrix form as follows A = [A 1 A A 3 ], B = B 1 B B 3, α 11 α 1 α 13 α = α 1 α α 3. α 31 α 3 α 33 The inner product of two vectors which is product of a bra and a ket, denoted by Dirac as A B but more commonly by omitting one of the middle lines as A B ] B 1 A B = [A 1 A A 3 = A 1 B 1 + A B + A 3 B 3. B B 3 There are a few important operations which we find very frequently when dealing with a quantum system. The product of a bra with a linear operator gives a row vector. ] α 11 α 1 α 13 A α = [A 1 A A 3 α 1 α α 3 = α 31 α 3 α 33 ] [A 1 α 11 + A α 1 + A 3 α 31 A 1 α 1 + A α + A 3 α 3 A 1 α 13 + A α 3 + A 3 α 33. Similarly product of a linear operator with a ket gives a column vector. α 11 α 1 α 13 B 1 α 11 B 1 + α 1 B + α 13 B 3 α B = α 1 α α 3 B = α 1 B 1 + α B + α 3 B 3. α 31 α 3 α 33 B 3 α 31 B 1 + α 3 A + α 33 B 3 7

19 The combined product of a bra, a linear operator and a ket gives a complex number. ] α 11 α 1 α 13 B 1 A α B = [A 1 A A 3 α 1 α α 3 B α 31 α 3 α 33 B 3 = (A 1 α 11 + A α 1 + A 3 α 31 )B 1 (A 1 α 1 + A α + A 3 α 3 )B (A 1 α 13 + A α 3 + A 3 α 33 )B 3. A ket times a bra generates a linear operator. B 1 ] B 1 A 1 B 1 A B 1 A 3 B A = B [A 1 A A 3 = B A 1 B A B A 3. B 3 B 3 A 1 B 3 A B 3 A 3 Any bra can be converted to a corresponding ket by taking the transpose of it and replacing each element by its complex conjugate. ] Ā 1 A = [A 1 A A 3 A =. Similar operation is performed on a linear operator α and the result is called the adjoint of α and it is denoted as α ᾱ 11 ᾱ 1 ᾱ 31 α = ᾱ 1 ᾱ ᾱ 3, ᾱ 13 ᾱ 3 ᾱ 33 where ᾱ 11 is the complex conjugate of α 11. Following identity is satisfied by the adjoints of two operators (αβ) = β α. Ā Ā Outer product As shown above, the product of a ket and a bra gives a matrix which acts as a linear operator. This kind of product is called outer product. The outer product φ ψ acting on a vector ϕ returns a new vector. ( φ ψ ) v = ψ v φ. A projector operator P i is i i i, where i is one of the basis in a basis set of a given space. Lets operate P i on a vector v = i v i i. Then, ( i i i ) v = i i i v = i i i v = i i v i = v. 8

20 So, we conclude that i i i = I. This is called completeness relation. The outer product notation of a matrix is A = = N e i e i A e j e j i,j=1 N e i A e j e i e j, i,j=1 where elements of matrix A is A ij = e i A e j. Following that, an identity matrix can also be written as N I = e i δ ij e j, i,j=1 where δ ij is kronecker delta function defined as 1 i = j δ ij =. 0 i j 1..5 Quantum Bit A bit is a basic unit of information used in computers. A classical bit can only have either of the two possible states(0 or 1) with definite probability. On the other hand a quantum bit has some non-unit probability attached to both the values. The qubit is represented as a state which is superposition of both. Mathematically, we can write the qubit state as a linear combination of both the states, as follows ψ = α 0 + β 1, where α and β are the probability amplitudes and α and β are the probabilities to get the states 0 and 1 respectively. Also, the total probability should be 1. α + β = Matrix and linear operators An operator is a function over the space of a quantum state which when act on one state produces another physical state with different properties. An operator can be written in matrix form to transform one basis vector into another. For example, if φ i and φ j are two basis vectors, they can be connected in the following way A ij = φ i  φ j, 9

21 where, A 11 A 1 A A 1n A Â = 1 A A 3... A n A n1 A n A n3... A nn Most operators in quantum mechanics are linear operators, which allows us to represent quantum mechanical operators as matrices and wave functions as vectors in some linear vector space Eigenvectors and eigenvalues We can find the eigenvalues and eigenvectors corresponding to an operator in matrix form in quantum mechanics. When an operator A acts on some vector ψ, we get a new vector ψ. But there can be a special case where the result could be the same vector as earlier or it could be a scalar multiple of the original vector. This can be written in the form of an equation as follows A ψ = a ψ. This can be written as (A ai) ψ = 0, where I represents the identity matrix of the same dimension as A. The solution exists when det(a ai) = 0. One of the postulates of quantum mechanics says that when measuring an observable of the system, the value obtained will be one of the eigenvalues of the operator corresponding to that observable. After the measurement is done, the state of the system will be corresponding eigenvector Inner products If f and g are two vectors such that f = f 1 î + f ĵ + f 3ˆk and g = g 1 î + g ĵ + g 3ˆk, Dot product of two vectors is defined as following f. g = f 1 g 1 + f g + f 3 g 3. Generalizing the dot product to n dimensions f. g = n f i g i. i=1 10

22 To find the magnitude of a vector we take dot product of the vector with itself and then take the square root of the number. So, the length of the vector f will be f = f. f = n fi. However, for complex vectors i.e. f i s being complex numbers, then f i will be a negative number and it will not define the magnitude of the vector properly. Inner product is used to generalize such calculation even for complex numbers. While calculating the inner product value, the first vector is complex conjugated. f g = i=1 n fi g i. i=1 If f i s are real, the inner product is same as the dot product. Using this definition of inner product, the length of a vector is always a positive number. f = f f = n f i. In quantum mechanics, the wave function and state vector can be used interchangeably. They are just two different ways of mathematically defining a quantum system and applying different operations on it. The length of a vector is similar to the norm of a function. A vector or a function is normalized if its norm is 1 i.e. f f = 1. Also, two vectors or two functions are orthogonal to each other if their inner product is 0 i.e. f g = 0. This type of vectors which are both normalized and mutually orthogonal occur a lot in quantum mechanics and are termed as orthonormal. i= Tensor products To explain many quantum mechanical phenomena, we need two or more particles together forming a system. We need to have a mathematical tool to describe such a system. So, if there are two particles, one described by v in vector space V and the other described by w in vector space W. We can represent the combined system by a notation (v, w) where the first item is the state of the first particle and the second item denotes the state of the second particle. But this notation does not give a general state of the two particle system just like how the individual particles states were defined. Here, tensor product notation plays an important role. We can define the state of the combined system as v w, which will be a new vector in the vector space V W that has the information of quantum states of the two particles. The operation is the tensor product. So, we have v w V W where v V and w W. If V is of m dimensions and W is of n dimensions, then the tensor product V W is of mn dimensions. If { v 1, v,..., v n } is the basis set of V and { w 1, w,..., w n } is the basis set of W, then the basis set of 11

23 tensor product V W is v i w j where i = 1,,..., m and j = 1,,..., n. In matrix form, the tensor product of two matrices can be calculated in the following way. If A and B are two matrices of dimension m and n respectively. A 11 A 1 A A 1m A A = 1 A A 3... A m A m1 A m A m3... A mm and B 11 B 1 B B 1n B B = 1 B B 3... B n B n1 B n B n3... B nn So, the tensor product A B is calculated as follows A 11 B A 11 B 1n A 1m B A 1m B 1n A 11 B n1... A 11 B nn A 1m B n1... A 1m B nn. A B = A m1 B A m1 B 1n A mm B A mm B 1n A m1 B n1... A m1 B nn A mm B n1... A mm B nn For two matrices A and B, det(a B) = (deta) m (detb) n, where det denotes determinant of the matrix. T r(a B) = (T r A)(T r B), where T r denotes the trace of the matrix. Following properties are to be satisfied when using tensors of two vectors. u v 1 + u v = u (v 1 + v ) u 1 v + u v = (u 1 + u ) v (αu) v = u (αv) = α(u v) A B( v w ) = A v B w, where α is a scalar and u, v, u 1, u, v 1, v are vectors. A and B are operator matrices. 1

24 1..10 Operators An operator in physics is a function over the space of physical states which act upon one state and delivering another state with different information as compared to the previous state. If we have a function f(x) and an operator Â, then Âf(x) is a new function φ(x). We can also use the matrix representation of such functions and it is associated with a measurable parameter for a physical system. There are some basic operators used in quantum mechanics like position operator, momentum operator and Hamiltonian operator denoted by ˆx, ˆp and Ĥ respectively. ˆx = x ˆp = ι Ĥ = ˆp m + ˆV (ˆx), where ˆp m is the kinetic operator and ˆV is the potential energy operator. So, we do notice here that an operator can be obtained by adding two operators. We can obtain a new operator by carrying out various algebraic operations. If we apply an operator Ĉ on a function f(x), where Ĉ =  ˆB Ĉf(x) =  ˆBf(x). The result will be the operator ˆB applied on f(x) giving intermediate result φ(x) and then operator  applied on φ(x) giving out the final result. Commutator The combination of operators of the form  ˆB ˆB is common in quantum mechanics. It signifies the order of applying the two operators on a system and it is termed as commutator. If the value of the commutator is 0, then the two operators are said to commute each others otherwise they do not commute. The short hand notation for commutator of  and ˆB is [Â, ˆB]. [Â, ˆB] =  ˆB ˆBÂ. Expectation Value Given a wavefunction ψ(x) and an operator O corresponding to some physical property of ψ(x), the expected value is defined as Q = + ψ (x)qψ(x)dx. Using dirac notation, where Q and ψ are in matrix representation. Q = ψ Q ψ. 13

25 A few kinds of operators are defined as follows, which are very important in quantum mechanics. Linear Operator An operator  is said to be linear if Â(cf(x)) = câf(x) and Â(f(x) + g(x)) = Âf(x) + Âg(x), where f(x) and g(x) are two appropriate functions and c is a constant. Hermitian Operator An operator  is called the hermitian conjugate of  if ( ψ) ψdx = An alternate name for hermitian conjugate is adjoint. The operator  is called hermitian if  =  (Âψ) ψdx = ψ Âψdx. ψ Âψdx. Unitary Operator Generally, when we use an operator to measure a quantum state, we obtain some information about the quantum state. At the same time, some information from the original state is destroyed. Unitary operation is another kind of operation which do not destroy any kind of information from the original system. One example is the rotation operator X which rotates a spin 180 degrees around x axis. X(a + b ) = a + b Bloch sphere The Bloch sphere provides the geometric representation of a quantum state as points on the surface of a unit sphere. Many operations used in quantum information processing tasks can be described using Bloch sphere. Let us try to derive how the Bloch sphere came into picture to represent quantum states. A pure state ψ in a dimensional quantum system can be written as following ψ = α 0 + β 1, where α and β are complex numbers. α and β are the probabilities that the state will be measured to be 0 or 1 respectively. When we normalize the wave function ψ, it puts on a condition on α and β that α + β = 1. 14

26 To understand the Bloch sphere representation of quantum states, we need to understand how the complex numbers are represented in polar coordinates. Lets say z is a complex number denoted by x + ιy represented pictorially as following in polar coordinates Figure 1.3 Polar Coordinates Representation of a complex number We can write x = rcosθ and y = rsinθ. So, z = r(cosθ + ιsinθ) and using Euler s Identity Therefore, we get e ιθ = cosθ + ιsinθ. z = re ιθ. Now, coming back to the qubit state representation, α and β are complex numbers. We can express the same state in polar coordinates as ψ = r α e ιφα 0 + r β e ιφ β 1. We see, there are four real parameters in this equation r α, r β, φ α and φ β. However, we can reduce the number of parameters using the fact that multiplying a complex number with an arbitrary factor e ι γ has no observable consequence on its magnitude. This can be proved as follows. e ιγ α = (e ιγ α) (e ιγ α) = (e ιγ α )(e ιγ α) = α α = α. So, we can multiply our state by e ιφα giving, ψ = r α 0 + r β e ιφ β ιφ α 1 = r α 0 + r β e ιφ 1. 15

27 Here, the number of parameters are reduced from four to three r α, r β and φ = φ β φ α. Going further, we also have the normalization constraint i.e. ψ ψ = 1 which simplifies to the sum of the magnitudes of the coefficients of 0 and 1 should be 1. Again using cartesian representation ψ can be written as, And imposing the normalization constraint, ψ = r α 0 + (x + ιy) 1. r α + x + ιy = r α + (x + ιy) (x ιy) = r α + (x ιy)(x ιy) = r α + x + y = 1. This is the equation of a unit 3D sphere with cartesian coordinates (x, y, r α ). We will move to polar coordinates again. Figure 1.4 Polar Coordinates Representation of a complex number in 3D Cartesian coordinates are related to polar coordinates as x = rsinθcosφ y = rsinθsinφ z = rcosθ. Coming back to the quantum state equation, renaming r α to z ψ = z 0 + (x + ιy) 1. 16

28 Remembering that r α, x and y make a sphere of unit radius, r in polar coordinates representation of (x, y, z) will be 1. ψ = cosθ 0 + sinθ(cosφ + ιsinφ) 1 = cosθ 0 + e ιφ sinθ 1 So, we have just two parameters defining the point on the bloch sphere. We will find out what should be the range of values of θ and φ. We notice that θ = 0 => ψ = 0 and θ = π => ψ = eιφ 1. Varying θ from 0 to π and φ from 0 to π we get all possible states of ψ and that covers the half of the sphere. Considering the states corresponding to the opposite points on the bloch sphere, which have polar coordinates as (1, π θ, φ + π). ψ = cos(π θ) 0 + e ι(φ+π) sin(π θ) 1 = cosθ 0 + e ιφ e ιπ sinθ 1 = cosθ 0 e ιφ sinθ 1 ψ = ψ. We see, the two opposite points on the bloch sphere differ only by a phase factor of 1. They are equivalent bloch sphere notations. So, to cover all the points of the sphere we can map where 0 θ π and 0 φ π θ = θ. Ψ = cos θ 0 + eιφ sin θ 1, The opposite points on the bloch sphere are orthogonal. Consider two qubit states ψ and Φ with polar coordinates (1, θ, φ) and (1, π θ, φ + π) respectively. Ψ = cos θ 0 + eιφ sin θ 1 The inner product of Φ and Ψ is, Φ = cos π θ = cos π θ Φ Ψ = cos θ cosπ θ = cos π = e ι(φ+π) sin π θ 0 e ιφ sin π θ 1. 1 sin θ sinπ θ Hence, the two opposite points representing states Φ and Ψ are orthogonal. 17

29 1..1 State vector The state of a physical system is represented by a state vector or ket and is denoted by the symbol ψ as suggested by Dirac. Because of the superposition of vectors in quantum systems, a ket can be expressed as a linear combination of basis kets φ i and coefficients c i. ψ = i c i φ i, where c i is the probability amplitude and c i is the probability of finding the system in the state φ i. Also, the state vector ψ are usually normalized i.e. ψ ψ = 1. This leads to the result that c i c i = i i c i = 1. The some of the probabilities is equal to 1. The normalization condition simply expresses the fact that the system must be found in any combination of the basis states. Pure vs Mixed state The quantum state ψ discussed above is a pure state. Basically, it is a superposition of φ i s. However, a mixed state is a mixture of two or more states like ψ. The analogy of pure and mixed states in physical systems can be shown with an example. Suppose an experiment in which a closed basket has one ball which has possible colors of red and green. Whenever we try to measure its color, we find the result as red or green with equal probabilities. The other scenario is that there are two balls in a closed basket, one is red and the other green. Whenever we pick out one ball, there is simple classical probability of 1 to get either red or green. Here, the first case represents the pure state of the ball and the second case is corresponding to the mixed state. If ψ is a mixed state of ψ 1 and ψ, it can be written as ψ = c 1 ψ 1 + c ψ, where c 1 and c are the classical probabilities with which ψ can be found in the state ψ 1 or ψ respectively. We have also discussed before that the classical probabilities arise because of our ignorance. So, mixed state is actually incorporation of ignorance in the quantum system. The expression looks similar to the the pure state but there is a conceptual difference between the two. The pure state ψ is the superposition of other states but the mixed state is actually the statistical mixture of a few states. By superposition, we mean a state is in both a and b at the same time. However, if there is a statistical mixture of a and b, we say it is either a or b. 18

30 1..13 The density matrix A density matrix is a way to describe a quantum system in mixed state. It extends the tools of classical statistical mechanics to quantum domain. The density matrix is also called a density operator. If the state ψ is a mixture of n states ψ i s with respective probabilities p i s, then the density operator ρ of the system can be written as follows ρ = The sum of p i s is unity. n ψ i ψ i p i. i=1 n p i = 1. i=1 Though ψ i s are normalized, they need not be orthogonal to each other. A pure state is just a special case of mixed state where n = 1. The density matrix for a pure state is ρ = ψ ψ. Properties of a density matrix Positivity : ρ 0 ρ is hermitian i.e. ρ = ρ The states are normalized i.e. T r(ρ) = 1 T r(ρ = 1 for pure state ) = < 1 for mixed state. The density operator can be written in the form of its matrix elements as follows ˆρ = u m ρ mn u n, mn where u i is complete basis set. Inversely, the elements ρ mn can be written in terms of ˆρ ρ mn = i p i u m ψ i ψ i u n. The matrix element, basically shows the correlation between u m and u n basis vectors. So, u m ψ i is the probability that ψ i is measured to be u m. Similarly, ψ i u n is the probability that ψ i is measured to be u n and p i is the probability that the system appears in the state ψ i. So, the whole term gives a correlation factor between u m and u n. Now, substituting p i ψ i ψ i = ˆρ ρ mn = u m ˆρ u n. 19

31 1..14 Trace Trace plays an important role in the calculations related to a quantum mechanical system. The trace of a matrix is nothing but the summation of all of its diagonal elements. If i is an orthonormal basis set for the Hilbert Space of the system, then the trace of an operator A is generally denoted as T r(a) and given by T r(â) = i  i. i If i s are also the eigenvectors of A and a i s are the corresponding eigenvalues. T r(â) = i = i = i i a i i a i i i a i. Then, the trace of the operator is just the summation of its eigenvalues. For any two operators  and ˆB on a Hilbert space H T r(â ˆB) = T r( ˆBÂ). T r(â ˆB) = = = = = = = = N φ n  ˆB φ n n=1 N φ n ÂI ˆB φ n n=1 N N φ n  ψ k ψ k ˆB φ n n=1 N k=1 n=1 N k=1 n=1 k=1 N φ n  ψ k ψ k ˆB φ n N ψ k ˆB φ n φ n  ψ k N ψ k ˆB k=1 N φ n φ n  ψ k n=1 N ψ k ˆBI ψ k k=1 N ψ k ˆB ψ k k=1 = T r( ˆBÂ). 0

32 Also, for Unitary Operator U, UU = I. Using above theory T r(uâu ) = T r(âu U). So, T r(uâu ) = T r(â). The expectation value of an observable  measured on a system in quantum state ψ can be written using trace.  = ψ  ψ = ψ i i  i i ψ i i = i  i i ψ ψ i i = i i i  ψ ψ i = T r(â ψ ψ ).s The reduced density operator and partial trace If  and ˆB are two density matrices corresponding to two individual systems. The joint density matrix of the composite system which consists of  and ˆB is denoted by ˆρ AB. Then, the subsystems are described by taking the partial trace of ˆρ AB. This is called reduced density operator. When we take the partial trace with respect to ˆB, the state obtained is corresponding to the system A, ˆρ a. Similarly, for getting the density matrix of B, we take the partial trace w.r.t. A. If ˆρ AB is a product state of A and B i.e. ˆρ AB =  ˆB. Then ˆρ a = T r B ˆρ AB = ÂT r( ˆB) ˆρ b = T r A ˆρ AB = ˆBT r(â).s Pauli Matrices Pauli Matrices are defined set of three matrices. They are, [ ] 1 0 σ 0 = σ i = 0 1 [ ] 0 1 σ 1 = σ x = 1 0 [ ] 0 ι σ = σ y = ι 0 [ ] 1 0 σ 3 = σ z =

33 Each of these three matrices are hermitian as well as unitary. The significance of these matrices arose from the SG Experiment in which the electrons were observed to quantized angular momenta. Each pauli matrix is an observable describing the spin of the electrons or the angular momentum in each of the three spatial directions. σ 0 is just an identity operator which does not change the state of the qubit when operated upon. The possible eigenvalues of these matrices are ±1 giving the number of possible outcomes after the measurement of the spins to be in each direction. Also, together with the identity matrices, pauli matrices can be used to for a basis for all matrices. 1.3 Postulates of Quantum Mechanics There are a few axioms or principles defined in quantum mechanics which are basically the ground rules for dealing with any quantum system and its evolution. These rules are called postulates of quantum mechanics. A few of them just form the mathematical background of quantum mechanics and others define the measurement processes. The postulates are defined in the sections below First postulate - Quantum State When a system has a particular set of quantities or parameters which can be measured and remain constant for a finite period of time, then we talk about the state of the system. The understanding of a state in classical mechanics is much different from that of the quantum mechanics. The classical states have a fixed specification but the knowledge of quantum states is with certain probabilities. We have discussed the concept of pure and mixed states. The pure state has the maximum specification as compared to any of the mixed state. The state of a quantum system is defined by a function Ψ(r, t) that depends on the position and time. For a single particle system r is a set of coordinates of that particle r = (x, y, z). For more than one particle, r is used to represent the complete set of coordinates i.e. (x 1, y 1, z 1, x, y, z,..., x n, y n, z n ). This function is also called the state function or the wave function. Max Born interpreted the wavefunction as the probability amplitude. We have the normalization condition which states that the particle has to be somewhere in the universe. In other terms, the probability of finding it is 1. Given a probability amplitude Ψ, the probability density is defined as Ψ Ψ. Multiplying the probability density with the volume element dτ gives the probability of finding the particle in the volume dτ. For a one particle system dτ = dxdydz. For multi-particle system dτ = dx 1 dy 1 dz 1...dx n dy n dz n.

34 Integrating the probability all over the space gives the total probability. Ψ(r, t) Ψ(r, t)dτ = 1. The wave function has to be single valued, continuous and finite Second Postulate - Hilbert space If H 1 is the Hilbert space associated with the system S 1 and H is the Hilbert space corresponding to the physical system S, then the composite system will have the Hilbert space which will be the tensor product of the two i.e. H 1 H. If ψ is a state in the space H 1 and φ is a state in H, the composite state will be ψ φ. Ψ = ψ φ. This expression says that the system S 1 and S are existing at the same time. This is different from the linear superposition principle, in which a particular qubit can have one of the two possible states with some probabilities. In the tensor product representation two different qubits are having two states at the same time. Consider the following composite state Ψ = 1 ( ψ 1 φ 1 + ψ φ ). Here, S 1 and S are not in a definite state, nor the composite state Ψ. We can say Ψ is an entangled state. The separability and entanglement will be discussed in later sections of this thesis. Such entangled state also gives rise to an important phenomena for quantum computation quantum non-locality, in which the two systems are correlated. The measurement done on the first qubit decides the output of the measurement on the second system for the same observable Third Postulate - Operators Every observable of a physical system is associated with a Hermitian operator allowing a complete set of eigenfunctions. An observable is a physical quantity like position, momentum, angular momentum, energy etc, which can be measured by an experimental procedure. When we apply an operator O on a function f(x) it produces another function g(x). Of(x) = g(x). The physical equivalent of it is that an operator acting on a quantum state, gives a new state different from the original one. In Dirac notation we can write, A ψ = φ, 3

35 where A is the operator. For particular cases φ can be written as aψ, where a is a real number. A ψ = a ψ. This is the eigenvalue equation of the operator A, ψ is the eigenfunction and a is the eigenvalue Linear Operator : A is a linear operator if A( ψ + φ ) = A ψ + A φ. In quantum mechanics, we deal with the linear operators mostly. So, most of the times we use the terms operators and linear operators interchangeably. The general problem in quantum mechanics is to solve the eigenvalue equation. The solutions i.e. the possible values of a can be discrete numbers or a continuous range of values. For example, the electron s spin can have 6 possible orientations. However, the position of a particle can take any value along the axis of real number Hermitian Operator : In the study of physical processes, we are interested in values of energy, angular momentum etc. The corresponding operators to measure these quantities, which are real numbers, are Hermitian operators. If A is a given operator, the corresponding adjoint operator is denoted by A. A is obtained by taking the transpose of A and also replacing each of the elements of A by its complex conjugates. A = (A T ) = (A ) T. where T represents the transpose operation and represents complex conjugate. For A to be hermitian A has to be self-adjoint. A = A. That is why, Hermitian operators are also called Self-adjoint operator. The eigenvalues of hermitian operators are real numbers. An adjoint operator A will satisfy the following equation ψ A φ = A ψ φ. Let A be a hermitian operator(a = A ) and ψ be one of its eigenvalues, the above equation can be written as ψ A ψ = A ψ ψ 4

36 With eigenvalue equations A ψ = a ψ and A ψ = a ψ, the above equation can be written as ψ a ψ = a ψ ψ a ψ ψ = a ψ ψ (a a ) ψ ψ = 0. ψ ψ 0. So, a a = 0 a = a. That implies, a, eigenvalue of A is real. If we have two operators A and B, their sum A + B is also an operator. (A + B)ψ = Aψ + Bψ. If we have two operators A and B, their product AB is also an operator. (AB)ψ = A(Bψ). If we have two operators A and B, then an operator can be defined as [A, B] = AB BA. If [A, B] = 0, then we say A and B commute and [A, B] is called commutator of A and B. If A is an operator and if there is an operator A 1 such that AA 1 = A 1 A = I. A 1 is the inverse operator and I is the identity operator Fourth postulate - Evolution The wavefunction of the system evolves in time according to the time-dependent Schrodinger equation. ĤΨ(r, t) = ι Ψ t, where Ĥ is a self-adjoint operator called Hamiltonian, ι is the imaginary unit and is the reduced Planck constant. Ĥ corresponds to the total energy of the system. Ĥ = m ˆ + Û(x). The first term is the kinetic energy operator. If r = (x, y, z), then ˆ = + +. In the x y z generalized terms, if Ψ(t) is the probability amplitude of a quantum state at time t, then Ψ(t + δt) is its probability amplitude at a later time t + δt so that Ψ(t + δt) = U(t + δt, t) Ψ(t), where U is a Unitary linear operator. The condition for unitary operator is UU = U U = I. 5

37 1.3.5 Fifth postulate - Expectation value If a system is described by the wavefunction Ψ, which is not an eigenfunction of an operator Â, then a distribution of measured values will be obtained. The average value of the observable property is given by A = Ψ ÂΨdτ Ψ Ψdτ, where the integration is over all the coordinates involved in the system. A is also called the expectation value, which is just the average of many measurements. If the wavefunction is normalized, then the denominator becomes 1. In Dirac notation, A = Ψ A Ψ. If we use the density matrix representation of a state, ρ, the expectation value can be obtained using trace. A = tr(aρ). 1.4 Quantum Information and Entropy Information provides a means of understanding and interpreting quantum theories. We need to quantify the information hidden in the state of a physical system for storage and communication. Entropy quantifies the amount of uncertainty involved in the measurement of a variable. Entropy and information are inversely related. The more uncertainty is there, the more information we have. For example, when identifying the outcome after the flip of a coin, we receive less information as compared to identifying which side we got in a six-sided dice. So, the more probable events are less informative. The rare events provide more information when measured. A unit of quantum information is the qubit which can take continuous values unlike classical bit which has discrete values. This happens because of the quantum mechanical phenomena superposition. There are different definitions for measuring entropy have been given. Two important definitions Shannon entropy and Von Neumann entropy are discussed below Shannon Entropy Shannon defined the information as the negative of logarithm of probability distribution. Since the uncertainty of an event and the information obtained are proportional, Shannon defined entropy for a discrete random variable X with possible values {x 1, x,..., x n } and probability mass function P (X) H(X) = n n P (x i )I(x i ) = P (x i )log b P (x i ), i=1 i=1 6

38 where b is the base of the logarithm used. When b =, the unit of entropy is commonly referred as bits. I(X) is the information content of X, defined as log b P (X). The logarithm of the probability density is useful because the information content of two independent systems will be added. So, H(X) is nothing but the expectation value of the information content of a random variable X. If the probability distribution is continuous rather than discrete, the summation is replaced by the integral. H(X) = P (x)i(x)dx = P (x)log b P (x)dx. So, the Shannon entropy quantifies how much information is conveyed on the average by a letter drawn from the ensemble X and gives how many bits are required to encode that information. The conditional entropy is the expectation value of the conditional probability distribution. For two random variables X and Y, the conditional p(x y) can be written using Bayes rule The conditional entropy H(X Y ) is p(x y) = p(y x)p(x). p(y) H(X Y ) = log p(x y) H(X Y ) = log p(x, y) log p(y) H(X Y ) = H(X, Y ) H(Y ). The mutual information I(X; Y ) quantifies how correlated the two messages are i.e. how much do we know about a message drawn from X as when we have read the message drawn from Y. I(X; Y ) = H(X) H(X Y ) = H(X) + H(Y ) H(X, Y ) = H(Y ) H(Y X) Von Neumann Entropy Given a density matrix ρ, Von Neumann entropy S(ρ) is defined as S(ρ) = tr(ρlnρ), where tr denotes the trace and ln denotes the natural matrix logarithm. This is an extension of Shannon entropy. If ρ is written in terms of eigenvectors 1,,..., n ρ = i α i i i. Then, the Von Neumann entropy becomes S = i α i ln α i. 7

39 Mathematical properties of S(ρ) Purity A pure state has S(ρ) = 0. In pure state ρ will have the form similar to ( ) 1 0 ρ =. 0 0 The entropy is 1log (1) + 0log (0) = 0. Invariance The Unitary change in basis does not change the entropy. S(U ρu) = S(ρ) because the entropy depends only in the eigenvalues of the density matrix and unitary transformation does not change the eigenvalues. Maximum If ρ has D non-zero eigenvalues, then S(ρ) log D. Equality holds when all eigenvalues are equal which corresponds to the maximum randomness. Concavity For λ i 0 and i λ i = 1, S( i λ i ρ i ) i λ i S(ρ i ). Which implies that the von neumann entropy is larger when we know less about how the state was prepared. Entropy of Measurement If we measure an observable A = u a u a u a u in the state ρ, the outcome a u occurs with probability p(a u ) = a u ρ a u. The Shannon entropy H(Y ) of the ensemble of measurement outcomes Y = {a y, p(a y )} satisfies H(Y ) S(ρ). The equality holds when A and ρ commutes where the randomness of the measurement outcomes is minimized. 8

40 Entropy of Preparation If a pure state is drawn randomly from the ensemble { ψ x, p x }, forming a mixed state with density matrix ρ = p x ψ x ψ x. x Then H(X) S(ρ). Equality holds when the states ψ x are orthogonal. This implies that the distinguishability is lost when we mix non-orthogonal pure states. Subadditivity For a bipartite state AB in the state ρ AB ρ AB S(ρ A ) + S(ρ B ), where ρ A = tr B ρ AB and ρ B = tr A ρ AB. Equality holds for ρ AB = ρ A ρ B i.e. when there is no correlation between the two states. 1.5 Quantum Correlation Correlation plays a prominent role in information technology. It helps in understanding the key differences between classical and quantum mechanics. One of the most counterintuitive features of quantum mechanics is its non-local nature, which makes a fundamental difference from the classical mechanics. Quantum mechanics allows correlations between the measurements performed at the spatially separated locations. This quantum behaviour can be exploited for various information processing tasks along with the limitations put by the quantum laws. To quantify the quantum correlations, two main types of quantities have been used : entanglement and discord. Entangled states display strong correlations that are impossible in classical mechanics. In such quantum systems, a local measurement can affect the system globally. When the two subsystem are not entangled even then correlation can exist between the two. Discord helps in understanding quantum-classical difference of correlation. Quantum discord measures nonclassical correlation between two subsystems of a quantum system. It includes correlations that are due to quantum physical effects but do not necessarily involve quantum entanglement [91]. Below, we discuss the the condition for separability and entanglement Entanglement and Separability condition Let H 1 and H are finite dimensional Hilbert spaces with basis sets { a i } n i=1 and { b j } m j=1 respectively. Using the postulates of quantum mechanics, the composite state space of the two systems will be 9

41 H 1 H with base states { a i b j } n,m i=1,j=1, also written as { a ib j }. A pure state of this composite system can be written as ψ = i,j c i,j ( a i b j ) = i,j a i b j. If a pure state ψ can be written in the form ψ = ψ 1 ψ where ψ 1 and ψ are the pure states, it is said to be separable. A mixed state described by ρ = k p k ρ k 1 ρ k, where k p k = 1 and p k 0, is separable. The sets {ρ k 1 } and {ρk } are mixed states of the respective subsystems. ρ k 1 and ρk are pure states. If we consider more than subsystems, lets say n subsystems with Hilbert spaces H = H 1 H... H n. A pure state ψ H is separable if it has the form ψ = ψ 1... ψ n. Similarly, a mixed state ρ is separable if it has the form ρ = k p k ρ k 1... ρ k n. The problem of deciding whether a state is separable is called the separability problem in quantum information theory. It is considered a difficult problem and has been shown to be NP-hard. A separability criterion is a necessary condition for a state to be separable. For the low-dimensional systems ( and 3), the Peres Horodecki Criterion is necessary and sufficient condition for separability [83]. Transposition Map of a density Matrix If p i are the eigenvalues and ψ i are the eigenvectors corresponding to a density operator ˆρ where p i 0 and i p i = 1, then the density matrix ˆρ can be written as ˆρ = i p i ψ i ψ i. ˆρ is a positive operator i.e. ˆρ 0. The matrix element ˆρ mn is ˆρ mn = m ˆρ n, where { n } is the orthonormal basis set. The transposition map of ˆρ is ˆρ T. The matrix element of ˆρ T is ˆρ T mn = ˆρ nm = n ˆρ m. ˆρ T is also a hermitian positive matrix. 30

42 For a bipartite system, if { n } and { v } are the orthonormal basis set for subsystems with Hilbert space H 1 and H respectively, then the matrix element of an arbitrary state ρ H = H 1 H in the product basis { n v } is ˆρ mu,nv = m u ˆρ n v. Taking transposition on the first subsystem ˆρ T mu,nv = ˆρ nu,mv = n u ˆρ m v. ˆρ T need not be a Hermitian positive matrix. However, for a separable state ρ = j p jρ 1 j ρ j, the partial transposed state ρ T 1 = j (ρ1 j )T ρ j is a positive operator Peres-Horodecki Criterion The necessary and sufficient condition for the state ˆρ of two qubits is inseparable is that at least one of the eigenvalues of the partially transposed matrix ˆρ T mu,nv is negative. The equivalent condition is given in the paper by Adhikari et. al. that at least one of the two determinants ρ 00,00 ρ 01,00 ρ 00,10 W 3 = ρ 00,01 ρ 01,01 ρ 00,11 (1.1) ρ 10,00 ρ 11,00 ρ 10,10 ρ 00,00 ρ 01,00 ρ 00,10 ρ 01,10 W 4 = ρ 00,01 ρ 01,01 ρ 00,11 ρ 01,11 ρ 10,00 ρ 11,00 ρ 10,10 ρ (1.) 11,10 ρ 10,01 ρ 11,01 ρ 10,11 ρ 11,11 is negative and W = ρ 00,00 ρ 01,00 ρ 00,01 ρ 01,01 (1.3) is non-negative Entangled states Entanglement demonstrates a spooky phenomena in which measuring a property of one entangled particle can reveal the outcome of the same measurement done on the other particle. No matter how far away the particles are. For example, it is possible to prepare a quantum state having two particles such that the total angular momentum is 0. So, if one particle is measured to have spin-up orientation, we can predict the other particle will show spin-down orientation if measured. The measurement done on the first system seem to be instantaneously affecting the second system. But quantum entanglement does not allow the transmission of any classical information faster than light. Such experiments reject the principle of local realism, which is that information about the state of a system should only be mediated by interaction in its immediate surrounding. One possible explanation can be that the particles 31

43 had acquired its properties when they last encountered each other. In this explanation, the reality of the property was determined locally [45]. Examples : A few specific examples of entangled state are Bell states Φ + = 1 ( 0 A 0 B + 1 A 1 B ) (1.4) Φ = 1 ( 0 A 0 B 1 A 1 B ) (1.5) Ψ + = 1 ( 0 A 1 B + 1 A 0 B ) (1.6) Ψ = 1 ( 0 A 1 B 1 A 0 B ). (1.7) These are four maximally entangled two qubit states. Bell state measurement is used in quantum teleportation and super dense coding. Consider two observers Alice and Bob who are far apart having one of the bell states i.e. Ψ +. Both are performing measurement on their part of the entangled state. They will both get the result 0 or 1 with equal probability of 1. Also one more constraint will be satisfied that whenever Alice measures his qubit to be 0, Bob will measure his qubit to be 1 and vice-versa. Werner Like State ( ) 1 p ρ = I 1 I + p ψ ψ, 4 where ψ = α 00 + β 11 and 0 p 1 and α + β = 1. When α = β = 1, the state becomes Werner State [16]. Bell Diagonal State It is a mixture of four Bell States. ρ = p i Φ + Φ + + p x Ψ + Ψ + + p y Ψ Ψ + p z Φ Φ. (p i, p x, p y, p z ) can be permuted to any order by local unitary transformation. ρ is positive density matrix and 0 p i 1, 0 p x 1, 0 p y 1 and 0 p z 1 [1]. 3

44 1.6 Entanglement assisted information processing tasks With increase in the amount of available data throughout the world, it is necessary to find new ways to store and process information. The effort to minimize the size of the basic components used in computers has been very successful till now but it can t continue indefinitely. In conventional form bits are used to store the information in binary form i.e. 0 or 1. By exploiting the laws of quantum mechanics, we can use qubits which can take infinitely many states. Those will the arbitrary combination or superposition of 0 and 1. Along with the presence of qubit, entanglement is another strange property in quantum mechanics, which can make a powerful tool while dealing with information. Quantum information processing uses qubits as its basic information units. Entanglement plays a pivotal role in information processing tasks such as superdense coding [78], teleportation [9], remote-state preparation, secret-sharing [61, 89, 6, 76], key-generation [46, 94, 18, 3], entanglement swapping [0, 101], remoteentanglement distribution [87], broadcasting [6, 7, 14, 7] and many more [4, 55, 9, 8], which are not otherwise possible by classical means. We will discuss quantum teleportation and superdense coding in detail in the next subsection Quantum Teleportation When we think of entanglement as a resource for quantum information processing, quantum teleportation is referred as the fundamental information protocol. It is a way to transfer quantum information from one place to another reliably through a classical channel. This process uses entanglement as the key factor to be as quick and reliable as it is. Quantum teleportation of states has applications in quantum communication as well as in quantum computation mainly to connect different quantum computers. It provides a mechanism of moving a qubit from one place to another without actually transferring the object which holds that qubit but what actually moves is the bit through the classical channel. The process starts with sharing an entangled pair of qubits between the two parties and measuring an arbitrary state at one location and the same state is received at the other end by a set of predefined conditions [58]. Presenting the teleportation protocol more formally, we start with an unknown state α 0 + β 1, which we are trying to teleport. We also have an entangled state 1 ( ). The first qubit of this entangled state is with Alice and the second qubit is with Bob. The pre-shared entangled state can be one of the four bell states shown in equations 1.4 to 1.7. So, Alice has two qubits, one she has to teleport and the other is one of the entangled pairs. Bob has the other qubit of the entangled pair. The state to be teleported Ψ is Ψ C = α 0 C + β 1 C. (1.8) The entangled pair Φ + AB Φ + AB = 1 ( 0 A 0 B + 1 A 1 B ). (1.9) 33

45 The state of the three qubit system is the product state between the entangled pair AB and the state C. Ψ ABC = 1 (( 0 A 0 B + 1 A 1 B )) (α 0 C + β 1 C ). (1.10) Now, Alice measures the composite state of A and C in Bell Basis, entangling A and C while disentangling B. The basis set 00, 01, 10, 11 can be written in terms of bell states from equations 1.4 to = 1 ( Φ + + Φ ) (1.11) 0 1 = 1 ( Ψ + + Ψ ) (1.1) 1 0 = 1 ( Ψ + Ψ ) (1.13) 1 1 = 1 ( Φ + Φ ). (1.14) Now, rewriting the state Ψ ABC after multiplying out the states and replacing the basis by bell basis. Ψ ABC = 1 Φ+ AC (α 0 B + β 1 B ) + 1 Φ AC (α 0 B β 1 B ) + 1 Ψ+ AC (β 0 B + α 1 B ) + 1 Ψ AC (β 0 B α 1 B ). We can notice here that the state of the qubit corresponding to Bob can be obtained by applying pauli matrices as described in section at page 1. Ψ ABC = 1 Φ+ AC σ 0 Ψ B + 1 Ψ+ AC σ 1 Ψ B + 1 Ψ AC ισ Ψ B + 1 Φ AC σ 3 Ψ B. When Alice measures the Bell state, she will find one of the four states with probability 1 4 and at the same time Bob will find his qubit in one of the states σ i Ψ C where i = 0, 1,, 3. Now, to send Bob, the state of C, Alice does not need to send the possibly infinite amount of information contained in the coefficients α and β, which may be real numbers out to arbitrary precision. She just needs to send the information about one of the four bell basis, which she is using for measurement. That will be maximum of bits of classical information. After receiving the classical information, Bob applies the appropriate pauli operator to receive the original state, which Alice wanted to teleport. We see that the two parties 34

46 never exchanged the qubits and the desired qubit state was delivered to the intended person. Let s say Bob used the basis Ψ AC for her local measurement, the overall state becomes Ψ ABC = Φ AC σ 3 Ψ B (1.15) Now, Bob applies σ 3 to the disentangled qubit C. σ i = I, he is left with the overall state Ψ ABC = Ψ AC Ψ B. (1.16) Bob has, therefore changed the state of his qubit to Ψ B = α 0 B + β 1 B, (1.17) which is identical to the original state of qubit C as in equation 1.8. There is this mapping between the basis used by Alice and the gate(σ 0, σ 1, ισ, σ 3 ) used by Bob for their respective measurements. Therefore, Bob can use that mapping to find out the state. If Alice uses Φ + basis, then she can decide to send two bits {00} through the classical channels used. Whenever Bob gets {00} as classical information, he can decide to apply the gate σ 0, which is equivalent to applying no operation. Similarly, {01}, {10}, {11} can be mapped to Ψ +, Ψ, Φ respectively for Alice and σ 1, ισ, σ 3 respectively for Bob Superdense Coding Superdense coding is one of the remarkable information processing tasks. It is a procedure to allow someone to send two classical bits to another party using only one qubit of communication. Earlier, it was not possible to transfer more than one bit using a single qubit. In teleportation, we sent quantum information using classical channel. Here, we will be able to send larger than expected amount of classical information through a quantum channel. Suppose Alice would like to send classical information to Bob using qubit. Bob will receive the qubit and recover the classical information via measurement. Since, non-orthogonal quantum states cannot be distinguished reliably, Bob will be able to measure only the states 0 and 1. So, it seems only 1 bit of information can be delivered to Bob i.e. 0 or 1. This does not give any benefit of using qubit instead of bit. However, using superdense coding protocol, one can improve this efficiency and send bits of information through a qubit. Similar to the teleportation process, superdense coding also uses entanglement between the sender and the receiver in the form of Bell pairs to achieve this efficiency. The mathematical representation of this process can be viewed as the reversed process of quantum teleportation. The process starts with an entangled state Ψ i.e. one of the bell states shared between two parties Alice and Bob [40]. Ψ = 1 ( ). (1.18) 35

47 The first qubit corresponds to Alice s and the second qubit to Bob s. Alice wants to send the two bits of information i.e. {00, 01, 10, 11} to Bob using the part of the entangled pair of qubits possessed by her. Alice first performs a single qubit operation on her qubit which will change the overall entangled state depending on which message she wants to send. If she wants to send {00}, {01}, {10}, {11}, then she will have to apply the operations σ 0, σ 1, ισ, σ 3 respectively. The corresponding output states after this operation will be 00: Ψ = 1 (σ 0 0 A 0 B + σ 0 1 A 1 B ) = 1 ( ) = Φ + (1.19) 01: Ψ = 1 (σ 1 0 A 0 B + σ 1 1 A 1 B ) = 1 ( ) = Ψ + (1.0) 10: Ψ = 1 (σ 3 0 A 0 B + σ 3 1 A 1 B ) = 1 ( ) = Φ (1.1) 11: Ψ = 1 (ισ 0 A 0 B + ισ 1 A 1 B ) = 1 ( ) = Ψ. (1.) After the operation, Alice can encode the information about the current state of the qubit and send to Bob. Now, Bob has both the qubits in the required state as in the equations 1.19 to 1. depending on the message sent by Alice. Next, what Bob has to do is to apply CNOT gate on the second qubit and then Hadamard(H) gate on the first qubit to get the final state. CNOT gate : CNOT gate operates on a quantum register consisting of qubits. It flips the second qubit(target qubit) if and only if the first qubit(control qubit) is 1. 36

48 Hadamard gate(h) : It acts on the single qubit and maps 0 to and 1 to 0 1. So, for each of the states obtained after Alice s operation, Bob s operation will be 00 : 01 : 10 : 11 : 1 ( ) 1 ( ) 1 ( ) 1 ( ) CNOT 1 ( ) H 00 (1.3) CNOT 1 ( ) H 01 (1.4) CNOT 1 ( ) H 10 (1.5) CNOT 1 ( ) H 11. (1.6) Finally, measuring this state will give Bob the classical message sent by Alice. 1.7 Quantum cloning Cloning is a process that makes an exact copy of a given state without altering the original state in any way. We see that the classical information can be copied perfectly. We copy the files present in our computer system daily and we don t face any restriction. In Dirac notation, the process of the quantum cloning can described as follows U( ψ 0 M ) = ψ ψ M, (1.7) where U is the cloning operator, ψ is the state to be cloned, M is the cloning machine state and 0 is the blank state, where the cloned state will appear. M is the state of the machine after cloning. This is one of the essential features of information processing. But the perfect cloning is forbidden owing to the rules of quantum mechanics [41, 47]. People have worked on building quantum cloning machines which does imperfect cloning. The quality of the machine is defined in terms of fidelity, which quantifies the distance between the states of the original and the cloned one, the value of which lies between 0 and 1. The research has been going on to make the fidelity of cloning as close to 1 as possible. A universal quantum cloning can have fidelity of cloning as high as No Cloning Theorem Wootters and Zurek first came up with the No cloning theorem in 198. After that, researchers digged into further to extend and generalize this theorem and also gave new insight to boundaries of classical and quantum mechanics. The theorem proved that there is no operation U which will satisfy 37

49 the equation 1.7 for all states ψ. This has been proved usimg the linearity of quantum mechanics operator and also using no-signalling theorem. The same has been proved in the next chapter. Theorem 1 (No cloning). A unitary operator U can clone two quantum states ψ and φ φ 0 ψ 0 U φ φ (1.8) U ψ ψ (1.9) if φ ψ is 0 or No Broadcast Theorem The no-cloning theorem is normally stated and proven for pure states; the no-broadcast theorem generalizes this result to mixed states [79, 100]. The term broadcasting can be used in different perspectives like broadcasting of states, broadcasting of entanglement and more recently in broadcasting of quantum correlation. [71] [73]. Barnum et al. were the first to talk about the broadcasting of states where they showed that non-commuting mixed states do not meet the criteria of broadcasting. Many authors also investigated the problem of secretly broadcasting of three-qubit entangled state between two distant partners with universal quantum cloning machine and then the result is generalized to generate secret entanglement among three parties [38] [70, 37]. There are two main theorems which have been given to formalize the impossibility of broadcasting and its limitations [15] [31]. Theorem. A set of mixed states is broadcastable if and only if the states commute with each other. In 1996, Barnum et al. proved this theorem in their paper using the concept of fidelity between two density operators. Also, correlations in a single bipartite state can be locally broadcast if and only if the states are classical in nature [56] [3]. Later on, in 008, Piani et al. came up with a interesting results related to no broadcasting of quantum correlations. They narrowed down the possibility of broadcasting quantum correlation to a specific set of quantum bipartite states. Following definitions of kind of bipartite states were considered in their paper, given a bipartite state ρ. Definition 1. ρ is separable if it can be written as i p kσk A σb k, where p k is a probability distribution and each σ X k is a quantum state. Definition. ρ is entangled if it is not separable. Definition 3. ρ is classical-quantum if it can be written as i p i i i σ B i, where { i } is an orthonormal set, {p i } is a probability distribution and σ B i are quantum states. 38

50 Definition 4. ρ is classical-classical or strictly classically correlated if there are two orthonormal sets { i } and { j } such that ρ = ij p ij i j j j, with p ij a joint probability distribution for the indices (i, j). Theorem 3. Classical-classical states are the only states that can be locally broadcast. [84] 1.8 Complementarity In a quantum system two properties are called complementary to each other if the measurement of one affects the other. Resultantly, we can t measure both the quantities correctly at the same time. A few well known examples of this are the uncertainty principle, particle-wave duality [57], mind-body dichotomy etc. These were the preoperties considered by Niels Bohr who first introduced the concept of complementarity in 197 [96]. When we measure a physical property of a quantum system, three major physical systems are involved: 1) the system under examination ) the measuring devices used 3) the human body. There is a trade-off between precision by which each entity retains its state after the measurement is performed. 39

51 Chapter Cloning and Broadcasting Contents.1 No cloning theorem Probabilistic cloning Approximate cloning State Dependent cloner State Independent cloner Optimality of cloning machine To M copies - cloning machine Broadcasting No broadcast theorem Broadcasting of Entanglement Broadcasting via local cloning Broadcasting via non-local cloning Conclusion Quantum mechanics permits the cancellation of possibilities. -Nick Herbert 40

52 Abstract In this chapter, we will mainly discuss about the broadcasting of entanglement via local and non-local cloning. We will first see that the perfect cloning is not possible in quantum mechanics. We will discuss the proofs of no cloning theorem in detail. We also discuss different quantum cloning machines and how the quality of the cloned states depend on them. We use these cloning machines to broadcast the entanglement both locally and non-locally. In the local cloning process, we apply the cloning operator on each components of the system individually. On the other hand, the non-local broadcasting is obtained by applying a single quantum operator to all the components together. We also give the examples of broadcasting entanglement using different types of states like werner-like and bell-diagonal states for both the cases of local and non-local. We give the broadcasting range for the most general two-qubits state both by local cloning as well as non-local cloning. We also find out that the specified range is broader in case of non local cloning as compared to the local one. We also extend the study with cloning more than copies. We find out that, in case of local cloning broadcasting of entanglement is not possible for more than copies. However, using non-local cloning, we can create till 5 copies. 41

53 The term cloning is used in different contexts such as biological cloning and in copying information both in classical as well as quantum world. When cloning an organism, every single bit of their DNA has to be copied to another being. Though it is not ethical but it has been possible to make a clone of a human being. In classical computer, the information is encoded in terms of bits having possible values 0 or 1. When we copy a file in classical computers from one location to another, the exact same file is received without loss of any information. However, the quantum cloning shows a bit of deviation from the notion of perfect cloning [49, 67]. This has intrigued many researchers to pursue the related studies. In quantum cryptography, the impossibility of copying the information helps in preventing eavesdropping. People have also put efforts in building different types of cloning machines which could help in cloning approximately or probabilistically [1, 10]. First, we discuss why the perfect cloning is not possible and what are the limitations with a few theorems and proofs..1 No cloning theorem In quantum systems, the perfect cloning has simply been impossible. Given a state ψ, the cloning is the process of replicating it so that the new system becomes ψ ψ. Given an arbitrary state, we can t make an exact copy of it. When we measure a given state ψ for an observable, say A, what we get is an average value of that observable and also the original state is changed to one of the eigenvalues of A. So, we never know exactly what state ψ is in and it is not possible to reconstruct it. This process can be performed perfectly only for a set of orthogonal states. This is an intrinsic property of any quantum system, not that it is the consequence of human or apparatus errors. The very famous no signalling theorem i.e to communicate faster than light, also defies the possibility of perfect cloning. We discuss some theorems below which help understanding these behaviours. Theorem 4. An arbitrary state cannot be cloned perfectly. Proof. Let the state to be cloned is Ψ = α 0 + β 1. Let s say, U can clone any state perfectly Ψ 0 U Ψ Ψ (α 0 + β 1 ) 0 U (α 0 + β 1 )(α 0 + β 1 ) = α 00 + αβ 01 + αβ 10 + β 11. (.1) Since U is the linear operator as well U(α 0 + β 1 ) 0 = (αu 0 + βu 1 ) 0 = α 00 + β 11. (.) 4

54 The expressions.1 and. are the cloned state of the same original state but are not equal to each other always. The equality holds only if α = 1, β = 0 or β = 1, α = 0, which are the cases when Ψ = 0 or Ψ = 1. Hence, proved. proof using no-signalling condition. [90] No signalling condition says that it is not possible to send information faster than the speed of light. However, Nick Herbert proposed that the quantum correlation can used to communicate faster than light. Considering two parties: Alice and Bob at arbitrary distance sharing two qubits in an entangled state Ψ = 1 ( 0 A 1 B 1 A 0 B ). When Alice measures σ z, she finds her qubit in the eigenstates 0 or 1 with probability 1 each and consequently Bob finds his qubits in 1 respectively. Bob finds the mixed state to be = 1 I. When Alice measures σ x, her eigenstates are + and and Bob s eigenstates are and + respectively. The mixed state with Bob is = 1 I. Bob will never be able to know which basis has been used by Alice to perform measurement. Now, suppose Bob uses a perfect 1 cloning machine to clone whichever bit he receives after Alice s measurement. Now if Alice measures σ x, Bob s mixture is ρ x = If Alice measures σ z, Bob s mixture is ρ z = Since, ρ x ρ z, Bob will know with some probability which basis Alice has chosen by measuring his perfect clones. This is a violation of no-signalling condition, which came into picture when we assumed the perfect cloner on Bob s side. Hence, it is not possible to have a perfectly cloned arbitrary state. Theorem 5. A set of non-orthogonal states cannot be cloned perfectly. Proof. If U is a unitary operator, the inner product of two states before and after the operation will remain same. Suppose, ψ and φ are two states to be cloned to two blank states 0 and 0 using U. 0 0 = ψ φ ψ φ 0 0 = ψ φ ψ φ ψ φ (1 ψ φ ) = 0 ψ φ = 0 or 1. So, ψ and φ have to be orthogonal. Hence, proved. When we perform cloning, either we will get a cloned state with some error introduced as compared 43

55 to the original one or we will get the perfect copies but only few times after doing many such operations [69]. Duan and Guo differentiated these two types of quantum cloning as deterministic cloning and probabilistic cloning respectively. They constructed a probabilistic quantum cloning machine by a general unitary-reduction operator, which we discuss next.. Probabilistic cloning [44] In probabilistic cloning, the desired copies are formed only with certain probabilities. In this method, the cloning machine performs both unitary evolution and the measurement. Post measurement, the inaccurate copies are discarded. They also proved that only the set of linearly independent states can be cloned by the probabilistic quantum cloning machine and hence gave the following theorem. Theorem 6. The states secretly chosen from the set $ = { Ψ 1, Ψ,..., Ψ n } can be probabilistically cloned by a general unitary reduction operation if and only if Ψ 1, Ψ,..., Ψ n are linearly independent. The probabilistic cloning machine yields correct copies of the input state with certain non-zero probability and the incorrect copies are discarded. So, at least a few copies are the exact clones of the original state. They also derived the necessary conditions for possible success probability in PQCM and also the optimal success probability for 1 PQCM. The calculations performed by Duan-Guo can be summarized as follows. Given a set of states $ = { Ψ 1, Ψ,..., Ψ n }, the 1 cloning is described as U Ψ j x Σ y P (0) z = γ j Ψ j y P (j) z + 1 γ j Φ (j) xyz, (j = 1,,..., m), where U is an unitary operator, Σ is the initial blank state, P (0) and P (j) are normalized states, Φ (1),..., Φ (m) are m normalized states of composite system xyz and P (j) Φ (k) = 0. The subspace spanned by the states P (1),..., P (m) is denoted by H p. During the cloning process, after the unitary evolution U, a projection measurement is carried out on the system z. With the positive cloning efficiency γ j, the measurement projects the state of z into the subspace H p, then the state of x and y collapses to Ψ j x Ψ j y and the cloning is realized successfully. If we get other measurement projections with the probability 1 γ j, the cloning process fails. Following this, the proof for theorem 1 has been given in the paper. They also proved that the optimal cloning efficiencies of PQCM can be obtained from the positive semi-definite matrix X (1) ΓX z () Γ +, [ ] where X (1) = [ Ψ j Ψ k ], X z () = Ψ j Ψ k P (j) P (k) and Γ = Γ + = diag(γ 1, γ,...γ m ). 44

56 Finally, it has been shown that, the cloning efficiencies for cloning the two states set { Ψ 1, Ψ } satisfies γ 1 + γ Ψ 1 Ψ. Because of the more complex design and higher precision control, there are many experimental challenges involved in building the probabilistic cloning machine. In the past, as compared to probabilistic cloning machines, we can find more number of literatures related to demonstrating approximate cloning machines in various forms and varying qualities. In this thesis also, we have tried to improve the already existing approximate cloning machines for our specific purpose of broadcasting of entanglement..3 Approximate cloning As we have seen that no-cloning theorem forbids the perfect cloning of an arbitrary state. However, it never rules out the possibility of approximate cloning. In the deterministic way of quantum cloning machines, where we use unitary operators with no post measurement, we end up getting the approximate copies. The quality of these copies are defined by the distance between the original and the final states, normally termed as fidelity. Depending on whether the fidelity varies with the input state or not, the approximate cloning machine is termed as state dependent or state independent cloners. We discuss both the types of cloners with examples in further subsections. We have used Bures distance as a measure of distance between the original and the cloned state, defined as [4] d B (ˆρ 1, ˆρ ) = [ 1 T r ) (ˆρ 1 1 ˆρ ˆρ 1 1 ] 1 (.3).3.1 State Dependent cloner When the fidelity between original state and the cloned state depends on the original state, the cloner is said to be state dependent. Because of the No-cloning theorem, we know that an arbitrary unknown quantum state cannot be cloned perfectly. However, we can always try to copy the state with some non-unit fidelity. Wootters and Zurek considered a device that does clone only the basis set 0, 1 perfectly, which is an example of state dependent cloner. The description is as follows Wootters-Zurek cloner The cloning operations are given as follows [99]. 0 a A 0 a 0 b A 0, (.4) 1 a A 1 a 1 b A 1, (.5) 45

57 where A is the cloning apparatus used, A 0 and A 1 are the possible states of the cloning apparatus after cloning. This transformation is unitary and linear. When applying it to the state ψ = α 0 + β 1 leads to ψ A α 0 a 0 b A 0 + β 1 a 1 b A 1. If A 0 = A 1, the composite system of a and b, ψ ψ = α β 1 1, is a pure state which is not the desired output. If the output apparatus states are considered distinct and orthonormal to each other i.e. A 0 A 1 = 0, A 0 A 0 = A 1 A 1 = 1, the density operator of the system will be ˆρ abc = α 00A 0 abc 00A 0 + β 11A 1 abc 11A 1 +αβ 00A 0 abc 11A 1 + αβ 11A 1 abc 00A 0. The density operator of the system excluding machine state, after tracing out the apparatus is T r c ˆρ abc = ˆρ ab = α 00 ab 00 + β 11 ab 11. (.6) We can now calculate the individual density operators of a and b T r b ˆρ ab = ˆρ a = α 0 a 0 + β 1 a 1, (.7) T r a ˆρ ab = ˆρ b = α 0 b 0 + β 1 b 1. (.8) We can notice that the output states are identical but significantly different from the desired original density matrix, which is ˆρ in a = ψ a ψ = α 0 a 0 + αβ 0 a 1 + αβ 1 a 0 + β 1 a 1, (.9) where ˆρ in a stands for initial density matrix corresponding to a. The off-diagonal terms are eliminated in the output states Fidelity of cloning We can find out how close the output states are to the original state by overlap between the two quantum states. This is also termed as fidelity, F. F = a ψ ˆρ a ψ a = b ψ ˆρ b ψ b = α 4 + β 4 = α 4 + (1 α ). (.10) It depends on α which defines the original input state. When α = 1 or α = 0, which corresponds to the basis states 0 or 1 respectively, the fidelity is 1. This is consistent to the claim made earlier that the orthogonal sets of states can be cloned perfectly. The states with linear superposition of these orthogonal states are copied with non-unit fidelity. The maximally entangled state, the state with symmetric superposition α = 1 is copied with the worst fidelity F = 1. Buzek and Hillery analyzed the properties of the cloned states obtained after using Wootters and Zurek 46

58 cloning machine. They used Hilbert-Schimdt norm D to calculate the distance between input and output density matrices, let s say ˆρ 1 and ˆρ respectively. It is defined as D = ( ˆρ 1 ˆρ ). (.11) They first calculated the distance between the original state and the cloned state which is not exactly same but analogous to the fidelity, we calculated in equation.10. Also, it can be seen that ˆρ ab in equation.6 is an entangled state. To measure the degree of entanglement, Buzek and Hillery measured the distance between the state ˆρ ab and ˆρ a ˆρ b as given in equations.6,.7 and.8 using Hilbert-Schimdt norm from equation.11. They found that the worse the cloning fidelity is, better is the entanglement between states. They further investigated with a mixed state α 0 a a 0 + β 1 a a 1 instead of ˆρ in a in equation.9. They figured out that the Wootters and Zurek quantum cloning machine produces a strong entanglement between output states even for mixed initial state..3. State Independent cloner State independent cloner, also known as universal cloner, treats all input states equally well. Buzek and Hillery were the first to propose a universal cloning machine. Later this machine was also proved to be optimal [1]. For any input state ψ = α 0 + β 1, it would always yield the same optimal non-unit fidelity 5 6, independent of α. So, for maximally entangled state, where we got the fidelity of 1 with state independent cloner, we get better fidelity Buzek-Hillery cloner The transformation of basis states are as follows [5] 0 a A 0 a 0 b A 0 + [ 0 a 1 b + 1 a 0 b ] B 0, (.1) 1 a A 1 a 1 b A 1 + [ 0 a 1 b + 1 a 0 b ] B 1, (.13) where A i and B i are machine states after transformation. They considered the unitarity of transformation because of which the following relations hold A i A i + B i B i = 1; i = 0, 1; (.14) B 0 B 1 = B 1 B 0 = 0. (.15) Also, since machine states will not be of any significance to us and we will need to trace them out to get the actual state, we can assume the machine-state vectors are orthogonal to each other. A few more relations we can have A i B i = 0; i = 0, 1; (.16) 47

59 and A 0 A 1 = A 1 A 0 = 0. (.17) Again the input state is same as in equation.9. The degree of distortion after cloning is quantified using Hilbert-Schmidt norm (eq.11). To find the value of the unknown parameters in the cloner two conditions were imposed to get a universal cloning machine. The distortion, distance between the input and the cloned state, should be independent of the input state or α. The distance between ˆρ cl ab, cloned composite state and ˆρin a ˆρ in b, product state of a and b s input states, is independent of α. After solving these equations, the universal quantum cloning machine can be written as 1 0 a A , (.18) a A , (.19) 3 3 where + = and = as two orthonormal basis states and. and copying machine states after cloning can be expressed.4 Optimality of cloning machine After the two types of cloning machines given by Wootters and Zurek, 198 and Buzek and Hillery, 1996, Gissin and Masar in 1997, came up with a quantum cloning machine that could transform N identical qubits into M > N identical qubits. They also proved that the fidelity of these copies were optimal. The same has been proved in theorem 7. In this thesis, we have only used the cloner to clone only one state into multiple states i.e. N = 1. Theorem 7. There is an upper bound on the quality of a quantum cloning machine compatible with no-signalling theorem. [53] Proof. Let s say the initial density matrix in generalized form is ρ in ( m) = I + m σ If the density matrix of the qubits system after cloning is denoted as ρ out ( m), the density matrix corresponding to each of the qubits is T r 1 (ρ out ( m)) = T r (ρ out ( m)) = I + η m σ 48

60 where η is the shrinking factor of the bloch vector m. The most general output state can be written as ρ out ( m) = 1 (I + η( m σ I + I m σ) + 4 j,k=x,y,z t jk σ j σ k ) The universal cloning machine acts similarly on all input states. Assuming m is in z-direction, this gives the relations t xx = t yy, t xy = t yx and t xz = t zx = t yz = t zy = 0 and ρ out ( ) = 1 4 (I + η(σ z I + I σ z ) +t zz σ z σ z + t xx (σ x σ x + σ y σ y ) + t x y(σ x σ y σ y σ x )) 1 + η + t zz = t zz t xx + ιt xy t xx ιt xy 1 t zz η + t zz Now, using the no-signalling condition that the basis used for measurement should not matter for a distant observer, the condition applied is ρ out ( ) + ρ out ( ) = ρ out ( ) + ρ out ( ) This implies that t xx = t yy = t zz = t Now we will apply the condition of positivity on ρ out ( ). The eigenvalues are 1 4 (1 ± η + t) and 1 4 (1 t ± t + t xy). For the eigenvalues to be non-negative, t xy = 0 and t = 1 3 and η max = 3. We see that η can t be 1. Hence, perfect cloning is not possible. The cloning fidelity is given by F max = T r(ρ in ( m)ρ out ( m)) = I + η max = 5 6 In this thesis, one of the things which we analyse is how many copies can be created from 1 qubit. Also, how it affects the fidelity and the range of states for which these cloning are possible. Next, we discuss N M cloning machines given in the past and also how we can use it for broadcasting of entanglement locally as well as non-locally(discussed in section.6). 49

61 To M copies - cloning machine Gisin and Massar, in 1997, presented a quantum cloning machine that transform N identical qubits into M > N identical copies. They also proved that the fidelity of these copies were optimal. This cloning machine can be reduced to BH cloning machine for N = 1 and M = Gisin-Massar cloning machine where For an arbitrary input state ψ, the cloner is mathematically formulated as [54] U 1,M ψ R = M 1 j=0 α j (M j)ψ, jψ R j (ψ), (.0) (M j) α j = M(M + 1). (.1) The universal cloner U 1,M clones 1 qubit to M qubits. Initially, there are M 1 blank states, where the cloned states appear later. R denotes the initial state of the cloning machine and R j (ψ) s are the orthonormal states of the cloning machine after the state is cloned. The notation (M j)ψ, jψ means a symmetric and normalized state of M qubits, where M j qubits are in the state ψ and j qubits in the state ψ. αj is the probability that there are j errors among the M output states. Deriving Buzek-Hillery cloner We can derive the Buzek-Hillery cloner (eq.18 and eq.19) from Gisin-Massar cloner (eq.0 for M =. When M =, α 0 = 3 and α 1 1 = 3. R j can be taken to be and since they should be orthogonal machine states. If we are cloning the state 0 where M =, for j = 0, the expression 0, 0 0 means qubits are in the state 0 and no qubit is in state 1. Similarly, for j = 1, the expression will be 1 0, 1 1, which means 1 qubit is in state 0 and 1 in state 1, the normalized state for which can be written as So, for cloning state 0, the equation.6 can be rewritten in simplified form as ( ) U 1,M 0 R = 00 +, (.) 3 3 which is similar to Buzek-Hillery cloner in equation.18. Similarly for cloning 1, the constants α 0 and α 1 remain the same along with the machine states R 0 and R 1. The only difference comes with the terms corresponding to the possible combination of cloned states. For j = 0, α 1 is the coefficient corresponding to 1 0, 1 1, which says 1 qubit is in 0 and 1 in 1, the normalized state for which is again, For j = 1, α 0 is the coefficient corresponding to 0 0, 1, which means both the qubits are in the state 1 i.e. the state is 11. The cloner for 1 is reduced to ( ) U 1,M 1 R = (.3) 3 3 This is again similar to equation.19 from Buzek-Hillery cloner. 50

62 .5 Broadcasting Broadcasting is the more generalized process of cloning. The term cloning is used for copying a pure state. However, broadcasting is used when the state to be copied is a mixed state and we need to deliver its copies to two or even more number of recipients..5.1 No broadcast theorem Because perfect cloning is not possible, we can t broadcast the quantum information with 100% accuracy for obvious reasons. No broadcast theorem is formally stated as : [] Given a single copy of a state, it is impossible to create a state such that one part of it is the same as the original state and the other part is also same as the original state i.e. given an initial state ρ 1, it is impossible to create a state ρ AB in a Hilbert space H A H B such that the partial trace T r A ρ AB = ρ 1 and T r B ρ AB = ρ 1. Entanglement has been a very intriguing feature in quantum mechanics and there are so many quantum processes which make use of it. This work is mainly about broadcasting entanglement rather than just broadcasting a state. So, if we have an entangled pair of states initially, we use cloning transformations to create two or more entangled pairs. In the next section, we discuss it in more details by broadcasting several example states..6 Broadcasting of Entanglement So far, we understand the importance of entanglement and the quantum state of two entangled qubits such as the bell states [7, 13]. We also saw in sections.3.1 and.3. that the cloning of a qubit results in two entangled qubits [5, 95]. The loss of the information was encoded to the correlation between two non-local qubits in terms of entanglement [70, 35]. The two entangled qubits held by two parties at a distance are useful for quantum information processing tasks such as quantum teleportation and superdense coding, discussed mainly in this thesis [63, 17]. Buzek et. al. have showed in their paper in 1997 that quantum state can be partially broadcasted with the help of local operations [7]. We try to multiply such entangled pairs by copying the individual qubits or by copying the entangled state of two qubits i.e. by cloning the qubits locally or non-locally respectively [77, 97]. Ghiu investigated the broadcasting of entanglement by using local 1 optimal universal asymmetric Pauli machines and showed that the inseparability is optimally broadcast when symmetric cloners are applied [51]. In other works, authors investigated the problem of secretly broadcasting of three-qubit entangled state between two distant partners with universal quantum cloning machine and then the result is generalized to generate secret entanglement among three parties [8] [68]. Various other works on broadcasting of entanglement depending on the types of QCMs were also done in the later period [33, 85]. The next two 51

63 subsections cover the examples for broadcasting a given state using both local and non-local cloning [60, 7]..6.1 Broadcasting via local cloning Figure.1 Pictorial representation of local cloning In this subsection, we deal with the problem of broadcasting of quantum entanglement by using local cloning transformation. As mentioned we start with a two qubit state ρ 1 shared between two parties A and B. The first qubit 1 belongs to the party A while the second qubit belongs to B. Each of them will now individually apply a local copying transformation, on their own qubit to produce the state ρ 134. Definition.1 An entangled state ρ 1 is said to be broadcast after the application of local cloning operation U 1 U, where U 1 and U are unitary operators, on the qubits 1 and respectively, if for some values of the input state parameters, the non-local output states between A and B ρ 14 = T r 3 [U 1 U (ρ 1 )] ρ 3 = T r 14 [U 1 U (ρ 1 )] are inseparable, (.4) whereas the local output states for each of two parties A and B ρ 13 = T r 4 [U 1 U (ρ 1 )], ρ 4 = T r 13 [U 1 U (ρ 1 )] are separable. (.5) The state ρ 1 given in eqn.9 is a general mixed state and is not going to be entangled for all values of the input state parameters. However, when we talk about broadcasting of entanglement it is only relevant when the initial state ρ 1 itself is entangled. The range of input state parameters for which broadcasting will be possible is always going to be a subset of the range of the input state parameters for which ρ 1 is entangled. 5

64 Now, we will give the expressions for 1 M cloners, where M = 1,. Then, we give the expression for the qubits most generalized state and its cloned form. We also find out the broadcasting range using peres horodecki theorem for the generalized state and a few example states like WL and BD. We will see that one cannot clone more than states using local cloners To M local cloner Using GM cloner given in equation.0, we have 1 M cloner for the basis states 0 and 1 as following U 1,M 0 R = U 1,M 1 R = M 1 j=0 M 1 j=0 α j (M j)0, j1 R j, (.6) α M 1 j (M 1 j)0, (j + 1)1 R j, (.7) where (M j) α j = M(M + 1) (.8) and R is the machine state before cloning R i and R j are the machine states after cloning. Now, we will use this generalized cloner for getting specific cloners where M =, 3. We won t need M > 3 because we won t find any state which can be broadcast using 1 3 cloner itself. Now, we will use this cloner to copy the following states and analyse the broadcasting of entanglement. We will also see how many copies can be created using local cloning Broadcasting qubits most general state The generalized expression for a qubits mixed state is given by [39] ρ 1 = I 4 + (x i σ i I + y i I σ i ) + t ij σ i σ j = { x, y, T }, (.9) 4 i=1 where x = [x 1, x, x 3 ] and y = [y 1, y, y 3 ] are the bloch vectors corresponding to Alice and Bob s qubits respectively. T = [t ij ] is the correlation matrix of dimension 3 3. σ i, i = 1,, 3 are Pauli Matrices and I, n =, 4 is an identity matrix of order n. We will be using this representation ρ 1 = ( x, y, T ) for all the density matrices now onwards. i,j=1 After performing the cloning operations as described in the equations.6 and.7 and tracing out the machine states, we get the cloned states to be { ρ 13 = 3 x, 3 x, 1 } { 3 I 3 and ρ 4 = 3 y, 3 y, 1 } 3 I 3, (.30) 53

65 { ρ 14 = ρ 3 = 3 x, 3 y, 4 } 9 T, (.31) where x and y are the bloch vectors of ρ 1 and I 3 is an identity matrix of dimension 3. Using the Peres-Horodecki criterion discussed in section 1.4.1, for the local states ρ 13 and ρ 4 to be separable, the condition is 0 x y 3 4 and x 1 + x 3 + x 3 and y 1 + y 3 + y 3. (.3) For the non-local diagonal states ρ 14 and ρ 3 to be inseparable, the condition is W 3 < 0 or W 4 < 0 and W 0, (.33) where W, W 3, W 4 can be derived using ρ 1 as per peres horodecki criterion discussed in section We will exemplify the broadcasting range with the mixed entangled states such as WL and BD states. Werner like states These states can be expressed as ρ w 1 = { x w, x w, T w}, (.34) where x w = { 0, 0, p(α β ) }, T w = diag(pαβ, pαβ, p) with the constraints 0 p 1 and 0 α 1. After cloning, the local output states are { ρ 13 = ρ 4 = 3 xw, 3 xw, 1 } 3 I 3, (.35) and the non-local states are ρ 14 = ρ 3 = { 3 xw, 3 xw, 4 9 T w}, (.36) where T w is the correlation matrix of the state ρ w 1. Finally applying the conditions of separability for local states and non-separability for the non-local states, the final range comes out to be 3 4 < p 1 and L < α < L +, (.37) where L ± = 1 16 (8 ± p p ). For p = 1, the WL state reduces to NME state. The range of α for NME state is 1 16 (8 39) < α < 1 16 (8 + 39). Next, we calculate the broadcasting range for Bell Diagonal(BD) state [1]. Bell Diagonal states The BD state is formally expressed as ρ b 1 = { 0, 0, T b }, (.38) 54

66 where bloch vectors 0 are null vectors and T b = diag(c 1, c, c 3 ). 1 c i 1, i = 1,, 3. After applying the local cloning operations and tracing out the machine states, the local states are { ρ 13 = ρ 4 = 0, 0, 1 } 3 I 3, (.39) and the non-local states are ρ 14 = ρ 3 = { 0, 0, 4 9 T b}, (.40) where T b is the correlation matrix of ρ 1. Now the condition for the local states to be separable and the non-local states to be inseparable is (4c 1 4c 4c 3 + 9)(4c 1 + 4c 4c 3 9)(4c 1 4c + 4c 3 9) (4c 1 + 4c + 4c 3 + 9) < 0 or (4c 3 + 9)((9 4c 3 ) 16(c 1 c ) ) < 0. (.41) Along with the condition that the eigenvalues of ρ 1 should be positive, which is 1 ] [1 + ( 1) x c 1 ( 1) (x+y) c + ( 1) y c 3 ; (x, y = 0, 1). (.4) Impossibility of cloning more than two copies When we apply 1 3 non-local cloner, we do not find any state for which broadcasting of entanglement is possible..6. Broadcasting via non-local cloning In this subsection, we reconsider the problem of broadcasting of entanglement, however this time we will use non-local cloning transformation. This situation is quite analogous to the previous case where we have used local cloning operations. Here, the basic idea is that the entire state ρ 1 (given in Eq. (.9)) is in the same place and we want to create more copies. In order to do that, we apply a global unitary operation U 1 to produce ρ 134. Definition. [7, 7]: An entangled state ρ 1 is said to be broadcast after the application of nonlocal cloning operation U 1 together on the qubits 1 and, if for some values of the input state parameters, the desired output states ρ 1 = T r 34 [U 1 (ρ 1 )], ρ 34 = T r 1 [U 1 (ρ 1 )] are inseparable, 55

67 Figure. Pictorial representation of non-local cloning and the remaining output states ρ 13 = T r 4 [U 1 (ρ 1 )], ρ 4 = T r 13 [U 1 (ρ 1 )] are separable. We could have chosen either the diagonal pairs ( ρ 14 & ρ 3 ) instead of choosing the pairs: ρ 1 & ρ 34 as our desired pairs in the above definition. However, we refrain ourselves from choosing the pairs ρ 13 & ρ 4 as the desired pairs [6]. Now, we go through the 1 non-local cloner given by Buzek and Hillery. Then we give the expression for 1 M non-local cloner as an extension of the GM 1 M cloner Buzek-Hillery 1 non-local cloner For M = i.e. to make copies of a given state, the B-H cloner is given as follows [6] Ψ i 00 X 1 5 Ψ i Ψ i X i + 10 j i ( Ψ i Ψ j + Ψ j Ψ i ) X j (.43) where Ψ 1 = 00, Ψ = 01, Ψ 3 = 10 and Ψ 4 = taken on left hand side is a blank state of two qubits. Each term represents a state of six qubits, two for the original state to be cloned, two for the blank state which will later become the cloned state and two for the machine state. X is the initial machine state and X 1, X, X 3 and X 4 are the machine states after cloning to M non-local cloner We generalize the Gissin-Massar equation to clone two qubits using one unitary operator. We will also show that the obtained equation reduces to Buzek-Hillery non-local cloner (equation.43) for M =. The proposed non-local cloner U 1,M is as follows U 1,M ψ i Σ (M 1) R = M 1 j=0 α j n j (M j) ψ i, j ψ l R jl, (.44) k=0 56

68 where ψ 1 = 00, ψ = 01, ψ 3 = 10 and ψ 4 = 11. Σ (M 1) are M 1 blank states. R is the initial machine state. R jl are the machine states after cloning. α j is the probability that there are j errors out of M cloned state. For example, while cloning 00, the error states are 01, 10, 11. (M j) ψ i, j ψ l represent the normalized state, which is the summation of all possible terms having M j states are in ψ i and j states are in ψ l where l i. Also, the machine states can be represented as R jl = (M 1 j) ψ i, j ψ l. (M j) nj α j = M 1 j=0 (M j) n, (.45) j where n j is the number of ways in which j states can be chosen out of available basis set { ψ 1, ψ, ψ 3, ψ 4 } and i l. Now, we will verify if this cloner reduces to BH 1 non-local cloner for M = Derivation of Buzek-Hillery non-local cloner Now we will try to reduce this to B-H non-local cloner for M =. For j = 0, n becomes 1 and for j = 1, n = 3 because 3 error states are possible for every state. Hence, α 0 = 5 and α 1 = 3 5. The first coefficient matches i.e. 5, the second coefficient is = 1 10, which also matches. The 1 factor 6 comes as a normalization constant with all six possible error terms qubits Most General State - broadcasting two copies After performing the non-local cloning operations as described in the equation.43 and tracing out the machine states, we get the local cloned states to be { 3 ρ 13 = 5 x, 3 5 x, 1 } { 3 5 I 3 and ρ 4 = 5 y, 3 5 y, 1 } 5 I 3, (.46) ρ 1 = ρ 34 = { 3 5 x, 3 5 y, 3 5 T }, (.47) where ρ 1 is the cloned state of ρ 1, x and y are the bloch vectors of ρ 1 and I 3 is an identity matrix of dimension 3. Using the Peres-Horodecki criterion discussed in section 1.4.1, for the local states ρ 13 and ρ 4 to be separable, the condition is W l 3 0 and W l 4 0 and W l 0, (.48) where W l 3, W l 4, W l can be constructed from ρ 13 and ρ 4 respectively. 0 x y 3 4 and x 1 + x 3 + x 3 and y 1 + y 3 + y 3. (.49) 57

69 For the non-local diagonal states ρ 14 and ρ 3 to be inseparable, the condition is W nl 3 < 0 or W nl 4 < 0 and W nl 0, (.50) where W nl, W nl 3, W nl 4 can be derived using ρ 1 as per peres horodecki criterion discussed in section We will exemplify the broadcasting range with the mixed entangled states such as WL and BD states. Werner like state After cloning, the local output states are { 3 ρ 13 = ρ 4 = 5 xw, 3 5 xw, 1 } 5 I 3, (.51) and the non-local states are ρ 1 = ρ 34 = { 3 5 xw, 3 5 xw, 3 5 T w}, (.5) where T w is the correlation matrix of the state ρ w 1. Finally applying the conditions of separability for local states and non-separability for the non-local states, the final range comes out to be 5 9 < p 1 and H < α < 1 + H +, (.53) where H ± = 1 1 (6 ± p p ). For p = 1, the WL state reduces to NME state. The range of α for NME state is 3 6 < α < Next, we calculate the broadcasting range for Bell Diagonal(BD) state. Bell Diagonal states After applying the non-local cloning operations and tracing out the machine states, the local states are { ρ 13 = ρ 4 = 0, 0, 1 } 5 I 3, (.54) and the non-local states are ρ 14 = ρ 3 = { 0, 0, 3 5 T b}, (.55) where T b is the correlation matrix of ρ 1. The local states come out to be separable always irrespective of the input state. Along with the conditions for the positivity of the density matrix from equation.6.1., the other conditions for the non-local states to be inseparable is (3c 1 3c 3c 3 + 5)(3c 1 + 3c 3c 3 5)(3c 1 3c + 3c 3 5) (3c 1 + 3c + 3c 3 + 5) < 0 or (3c 3 + 5)((5 3c 3 ) 9(c 1 c ) ) < 0. (.56) 58

70 .6..5 Increasing the number of copies Now, we extend our study of broadcasting entanglement for more than two copies. We give the broadcasting range for qubits most general state and along with the examples for the WL states and BD states. We observed that, in case of non-local cloning, the range of broadcasting for non-diagonal entries(1, 3 4) are broader. So, we will only consider only non-local non-diagonal states for broadcasting purpose. Considering x = x w, y = y w and T = T w for WLS and x = 0, y = 0 and T = T b for BD states, we analyse the following cloned states with different number of copies Broadcasting 3 copies The cloned states are given by ρ 1 = ρ 34 = { 7 15 x, 7 15 y, 7 15 T }, (.57) We consider the examples of WL states and BD states. Werner Like States The broadcasting range comes out to be 5 7 < p 1 and H < α < 1 + H +, (.58) ( where H ± = ± ( )) p p. Bell Diagonal states The broadcasting range is (7c 1 7c 7c )(7c 1 + 7c 7c 3 15)(7c 1 7c + 7c 3 15) (7c 1 + 7c + 7c ) < 0 or (7c )((15 7c 3 ) 49(c 1 c ) ) < 0. (.59) Broadcasting 4 copies The cloned states expressed as ρ 1 = ρ 34 = { 5 x, 5 y, 5 T }, (.60) Calculating the broadcasting range for states like WL and BD. Werner Like States The broadcasting range comes out to be 5 6 < p 1 and H < α < 1 + H +, (.61) 59

71 where H ± = 1 8 ( ) 4 ± p p. Bell Diagonal states The broadcasting range is (c 1 c c 3 + 5)(c 1 + c c 3 5)(c 1 c + c 3 5) (c 1 + c + c 3 + 5) < 0 or (c 3 + 5)((5 c 3 ) 4(c 1 c ) ) < 0. (.6) Broadcasting 5 copies The cloned states are given by ρ 1 = ρ 34 = Werner Like States The broadcasting range comes out to be { 9 5 x, 9 5 y, 9 5 T }, (.63) 5 7 < p 1 and H < α < 1 + H +, (.64) ( ) where H ± = ± p p. Bell Diagonal states The broadcasting range is (9c 1 9c 9c 3 + 5)(9c 1 + 9c 9c 3 5)(9c 1 9c + 9c 3 5) (9c 1 + 9c + 9c 3 + 5) < 0 or (9c 3 + 5)((5 9c 3 ) (9c 1 9c ) ) < 0. (.65) Broadcasting 6 copies When we apply 1 6 cloner, we do not find a valid range of states for which broadcasting of entanglement is possible. We observe the results for the number of copies that can be created matches with the results stated in [14]..7 Conclusion We conclude this chapter with several observations of the results. In this chapter, we proposed a 1 M non-local cloner for broadcasting entanglement. We used this cloner for cloning a generalized 60

72 two qubits mixed state and further calculated the broadcasting range for specific examples of werner like states and bell diagonal states. A few things are worth observing : The magnitude of bloch vectors corresponding to locally cloned state is greater than that of nonlocally cloned state. That means, the locally cloned state is closer to the original state as compared to non-locally cloned state. Since, each state is cloned individually in case of local cloning it gives the best possible cloned state possible. The range of states for which broadcasting of entanglement is possible, is larger for non-locally cloned states. In other words, the states for which broadcasting of entanglement is possible are closer to maximally entangled state in case of local cloning as compared to that of non-local cloning. Also, more number of copies can be cloned using non-local cloner. In local cloning, only copies can be created. However, in non-local cloning we can broadcast till 5 number of copies. Both the results are there because of the fact that, in local cloning, two machine states are used while in non-local cloning only one machine state is used. So, the distribution of entanglement from the original state is more in case of local cloning owing to the presence of an extra machine state. Resultantly, less entanglement is being copied to the useful states. In non-local cloning, as the number of copies increases from to 5, the range of states for broadcasting entanglement becomes shorter, which is as expected for obvious reasons. Basically, the possible states for broadcasting get closer to the maximally entangled state. 61

73 Chapter 3 Complementarity of Information Processing Tasks with Broadcasting Fidelity Contents 3.1 Complementarity in Quantum mechanics Complementarity in Quantum information Complementarity between broadcasting fidelity and Information Processing Tasks Complementarity for Two qubits generalized mixed states Complementarity in Broadcasting multiple copies One may say the eternal mystery of the world is its comprehensibility. -Einstein 6

74 Abstract Complementarity have been an intriguing feature of physical systems for a long time. In this chapter we establish a new kind of complementary relations in the frame work of quantum information processing tasks. In broadcasting of entanglement we create many pairs of less entangled states from a given entangled state both by local and non local cloning operations. These entangled states can be used in various information processing tasks like teleportation and superdense coding. Since these states are less entangled states it is quite intuitive that these states are not going to be as powerful resource as the initial states. In this work we study the usefulness of these states in tasks like teleportation and super dense coding. More precisely, we found out bounds of their capabilities in terms of several complementary relations involving fidelity of broadcasting. In principle we have considered general mixed as a resource also separately providing different examples like a) Werner like states, b) Bell diagonal states. Here we have used both local and non local cloning operations as means of broadcasting. We extend this result by obtaining bounds in form of complementary relations in a situation where we have used 1 M cloning transformations instead of 1 cloning transformations. 63

75 3.1 Complementarity in Quantum mechanics The complementarity principle was formulated by Neils Bohr [96] [19]. He considered a few examples and concluded that the complementary behaviour can be seen in physical systems due to its intrinsic limitations. The two complementary properties cannot be measured correctly and simultaneously. One common example is the particle-wave duality of quantum objects, because of which the object can t be fully described in terms of wave or particle [4]. Heisenberg s uncertainty principle [59] is also a very apt example of complementary behaviour, which says that there is a fundamental limit to the precision with which a pair of properties, position and momentum, can be measured. An electron can have greater accuracy of its position only by trading with the loss in the accuracy of its momentum. Heisenberg also gave a mathematical constraint on the error caused while measuring position and momentum and is given by x p ħ, where indicates standard deviation, x and p are position and linear momentum respectively ħ is the Planck s constant divided by π. There are also articles talking about complementarity in information theory [80] [81]. We will discuss it in the next section. 3. Complementarity in Quantum information Apart from a pair of observables to be measured in a system, the concept of information encoded in a state also provide examples of complementarity. In 003, Oppenheim et. al found out the complementary relationship between local and non-local information for some well defined processes [80] [81]. For two parties each possessing one qubit, local information will come from some physical operation on the individual qubits while non-local information will come from process like teleportation. They showed that one can access to the local information or the non-local information but not both. It is suggested that these complementarities represent an essential extension of Bohrs complementarity to complex (distributed) systems which are entangled. Later in 011, Fedrizzi et. al. explored the distribution of information in a comprehensive range of mixed states [48]. They found that entanglement can be increased by reducing the correlation between between two subsystems [75, 74]. Recently, in 015, Sazim et. al. showed a new kind of complementarity between two different physical processes such as approximate quantum cloning and the deleting. They also showed that the fidelity of the cloning and deleting processes is related to the correlations generated during the process. They illustrated the same with correlation measures such as geometric discord, entropic quantum discord and negativity(n). They found that the fidelity decreases with increase of correlation and that the total correlation change in the cloning and deleting is bound by the maximum value of measure of correlation [88]. 64

76 The earlier research motivates us to investigate about the complementarity relations involved in the cloning process of an entangled state and the information processing tasks such as teleportation and superdense coding. We talk about the complementarity relation between broadcasting fidelity and change in teleportation fidelity/superdense coding capacity. Also, we give a bound on the two quantities both for the local cloning as well as non-local cloning. We also extend the study by increasing the number of copies to be cloned. 3.3 Complementarity between broadcasting fidelity and Information Processing Tasks In this section we present the central idea of our work where we establish the complementary relations between the fidelity of broadcasting process with the decremental change in the information processing abilities as a consequence of creating lesser entangled pairs from an initial entangled resource. This entire process of broadcasting which is involved in creating more entangled states from a single entangled resource is done by local as well as non local cloning operations. In each of these cases we separately calculate the change in the teleportation fidelity and the superdense coding capacity of the entangled state to find that these values can not be arbitrary in nature. This change has a trade off with the broadcasting fidelity of the process. In fact the sum of these two quantities is a constant expression. Interestingly this shows us that each of these quantities are complementary in nature. In other words better is the fidelity of broadcasting, lesser is the change in the information processing capabilities. In short if we broadcast well we preserve the capacity of the entangled state to be used as a useful resource. We will try to analyse this with the most general qubit state, which is represented as ρ = 1 4 (I I + i x i.σ i I + i y i.i σ i + ij t ij σ i σ j ), (3.1) where x 1, x and x 3 are elements of bloch vector X. y 1, y and y 3 are elements of bloch vector Y with 0 X 1 and 0 Y 1. t ij, i, j = 1,, 3 are elements of the correlation matrix T. σ 1, σ, σ 3 are the pauli matrices and I is the Identity matrix. The qubits most general state can also be represented in the form of bloch vectors as { X, Y, T }. After applying the local cloner, the cloned state is { 3X, 3Y, 4 9 T }. Similarly, the state after applying the non-local cloner is { 3 5X, 3 5Y, 3 5T }. Next we discuss the information processing tasks such as Teleportation fidelity and Superdense-coding capacity before and after the cloning and their relationship with the Broadcasting fidelity. First, we define the three quantities. Broadcasting Fidelity: Broadcasting fidelity defines how close the cloned and the original states are. 65

77 The more the broadcasting fidelity value is, the better cloning has happened. Mathematically, it is given by F B(ρ 1, σ) = tr[ ρ1 σ ρ 1 ]. Teleportation Fidelity: Quantum teleportation is about sending a quantum information belonging to one party to a distant party with the help of a resource entangled state [58]. It is well known that all pure entangled states dimensions are useful for teleportation [11, 64]. However, the situation is not so trivial for mixed entangled states. There are entangled states which can not be used as a resource for teleportation [64]. However after suitable local operation and classical communication (LOCC) one can always convert them to a states useful for teleportation [30]. For a given two qubit mixed state as represented in Eqn[3.1] as a resource state we have the teleportation fidelity(t F ) as, T F = 1 [ ( i ui )] (3.) The quantities u i are the eigenvalues of the matrix U = T T. T is the correlation matrix as in the definition of qubits most general state. A quantum state is said to be useful for teleportation when the value of the quantity T F is more than the classically achievable limit of fidelity of teleportation, which is 3. The entangled Werner state [98, 16] in dimensions is one example of a useful resource for teleportation for a certain range of classical probability of mixing [64]. Other examples, of mixed entangled states as a resource for teleportation are also there [34]. Super Dense Coding Capacity Quantum super dense coding involves in sending of classical information from one sender to the receiver when they are sharing a quantum resource in form of an entangled state. More specifically superdense coding is a technique used in quantum information- theory to transmit classical information by sending quantum systems [40]. It is quite well known that if we have a maximally entangled state in H d H d as our resource, then we can send log d bits of classical information. In the asymptotic case, we know one can send log d + S(ρ) amount of bit when one considers non-maximally entangled state as resource [11, 6,, 3, 9]. It had been seen that the number of classical bits one can transmit using a nonmaximally entangled state in H d H d as a resource is (1 + p 0 d 1 ) log d, where p 0 is the smallest Schmidt coefficient. However, when the state is maximally entangled in its subspace then one can send up to log(d1) bits [36]. The superdense coding capacity for a two qubit mixed state is given by [87] where S(A) is the Von-Neumann entropy of the state A. d C = 1 + S(ρ ) + S(ρ 1 ), (3.3) 66

78 3.3.1 Complementarity for Two qubits generalized mixed states In this subsection we consider the most generalized two qubits mixed state ρ 1 as our initial resource. This fifteen parameter general state is given by Eqn[3.1]. Then we apply generalized Buzek-Hillery cloning transformation on this state to obtain four qubits state ρ 134 as output. The cloning is carried out both locally on individual qubits and non locally on both the qubits. Then in each case we trace out the redundant qubits to obtain the newly generated entangled pairs σ = ρ 14 / ρ 3 (for local cloning) and σ = ρ 1 / ρ 34 (for non local cloning). In each case we compute the broadcasting fidelity. In addition to that we also compute the change in the teleportation fidelity ( T F = T F (ρ 1 ) T F (σ)) and also the change in the super dense coding capacity ( DC = DC(ρ 1 ) DC(σ)) in the transition of the initial state ρ 1 to the final state σ in the decompression process. Interestingly we observe that the sum of both these quantities with the broadcasting fidelities is always bounded with two qubits resource we started with. We plot these sums in subsequent figures for both local and non local cloning transformations. In other words we show that these two quantities namely broadcasting fidelity and the change in information processing capabilities ( DC(T F )) are complementary to each other. Increase in one quantity will bring down the other. Figure 3.1 a) Plotting the sum DC + F B with T r[ρ ]. b) Plotting the sum T F + F B with T r[ρ ]. In each of the cases the cloning operation applied is local cloning. Figure 3. a) Plotting the sum DC + F B with T r[ρ ]. b) Plotting the sum T F + F B with T r[ρ ]. In each of the cases the cloning operation applied is non local cloning. In figure 3.1 we plot the sum a) DC + F B and b) T F + F B with the trace of the square of the initial state ρ. The randomly generated points are set of points on Bloch sphere, each point representing a two qubit state. This figure corresponds to the situation when we have used 67

79 local cloning transformation on respective qubits. In both of these cases the respective sums are bounded. In a) the sum of these quantities can never go beyond as F B is bounded by 1 while the teleportation fidelity can never go beyond 1. So, serves as an upper bound to this sum. Similarly in b) since the maximum dense coding capacity for a two qubit state is (for Bell states). So here 3 serves as an upper bound to this sum. Since for a given state the sum is always fixed the we conclude that that increase in any one of them will bring down the other. In figure 3. we once again plot the sum a) DC + F B and b) T F + F B with the trace of the square of the initial state ρ. However in this situation we have used non local cloning transformations for the purpose of broadcasting. Quite similar to the previous figure here we also observe that all these sums are respectively bounded by and 3. Next we exemplify this complementarity with the help of two well known mixed states namely a) Nonmaximally entangled states b) Werner-like States c) Bell diagonal States. In each of these cases we show that how broadcasting fidelity and the change in the information processing capabilities maintains a complementarity relationship with each other. Example A: Non-Maximally Entangled states Broadcasting by Local Cloning: Let s first discuss the simplest case of non-maximally entangled(nme) state to understand the complementary relation between broadcasting fidelity(fb) and T F / DC. The qubits NME states are given by ρ = ψ ψ, where ψ = α 00 + β 11, where α + β = 1. The expression for brodcasting fidelity is F B = 1 36 (5 16α β ). The plot for broadcasting fidelity vs α is given in figure 3.3. Figure 3.3 Plotting F B with α. The cloning operation applied is local cloning. 68

80 We observe that the broadcasting fidelity is minimum for the maximally entangled state(bell state with α = 1 ). Now we try to observe the relationship between broadcasting fidelity and change in teleportation fidelity(superdense coding capacity). The expression for teleportation fidelity of the initial state is T F i = (1 + αβ) 3 and that for the cloned state is T F c = 1 ( αβ). 54 Now the difference between the telportation fidelity of initial and after cloned states is T F = T F i T F c = 5 (1 + 4αβ). 54 Similarly, we find the expressions for superdense coding capacities before and after cloning and also the difference between the two. DC i = 1 α logα β logβ, ( DC f = 1 1 ] 48α ArcTanh[ 36 3 (β α ) DC = DC i DC f + 8 [ 4 ] 9 0α β ArcTanh 9 0α 13 β 6log [ 1 + 4α ] 30log [ 1 + 4β ] + 13log [ α β ] 36log3 + 10log5 (4α ( 3 ArcTanh [ β α ] + ArcTanh = [ 4 ] 9 0α β ArcTanh 9 0α 13 β [ ]) 3 (α β ) +6log [ 1 + 4α ] 36log [ 1 α ] 13log [ α β ] + 36log6 10log5 ) )., We plot broadcasting fidelity with change in teleportation fidelity and also with change in superdense coding capacity to demonstrate the complementary relation between both the quantities in figure 3.4. We can clearly see how the two quantities vary to each other in a complementary manner. Figure 3.5 shows how the quantities T F + F B vs α and DC + F B vs α. We can see how the bounds vary with states. Broadcasting by Non Local Cloning: 69

81 Figure 3.4 a) Plotting F B with T F. b) Plotting F B with DC In each of the cases the cloning operation applied is local cloning. Figure 3.5 a) Plotting F B + T F with α. b) Plotting F B + DC with α In each of the cases the cloning operation applied is local cloning. First, we find out the analytical expressions for broadcasting fidelity, teleportaion fidelity and superdense coding capacity before and after the cloning. F B = 7 10, T F i = (1 + αβ), 3 T F c = 1 (3 + αβ), 5 T F = 1 (1 + 4αβ), 15 DC i = 1 α logα β logβ, DC c = 1 ( 7log7 (1 + 3β )log(1 + 3β ) (1 + 3α )log(1 + 3α ) ), 10 DC = 1 ( 10 7log7 + (1 + 3β )log(1 + 3β ) + (1 + 3α )log(1 + 3α ) 10α logα 10β logβ ) 10 We plot the relation between broadcasting fidelity(fb) and α in figure 3.6. FB comes to be constant for all α. Figure 3.7 demonstrates how broadcasting fidelity(fb) and change in teleportation fidelity varies with α for non-local cloning. In this case we can see that even if F B remains same for all the states, T F and DC change. This happens because the ability to teleport or dense coding depends upon the state α. Figure 3.8 shows the bounds T F + F B and DC + F B in case of non-local cloning. 70

82 Figure 3.6 Plotting F B with α. The cloning operation applied is non-local cloning. Figure 3.7 a) Plotting F B with T F. b) Plotting F B with DC In each of the cases the cloning operation applied is non-local cloning. Figure 3.8 a) Plotting F B + T F with α. b) Plotting F B + DC with α In each of the cases the cloning operation applied is non-local cloning. Example B: Werner-like States The werner like states are given by where 0 p 1 and ψ = α 00 + β 11 and 0 α 1. ρ = p ψ ψ + 1 p I I, (3.4) 4 71

83 Broadcasting by Local Cloning: We get the mathematical expressions for the broadcasting fidelity using local cloner as ( F B = p + 14p + f 1 + f + ) f 1 f, where f 1 = p + 4p (7 8α + 8α 4 and f = p 5(3 + p) 16(14 + p(3 + 13p))α + 16(14 + p(3 + p))α 4 51p α p α 8. Teleportation fidelity for initial and cloned states and also the difference are stated below. T F i = 1 (3 + p + 4pαβ), 6 T F c = 1 (7 + 4p + 16pαβ), 54 T F = 5 (p + 4pαβ) 54 Similarly, calculating the superdense coding capacity for werner like states as ( DC i = 1 4 DC c = p(α β ) ArcTanh [ p(β α ) ] + 3(1 p)log(1 p) + (1 + 3p)log(1 + 3p) log(1 p (α β ) ) ( 36 + (9 4p)log ) ( 1 6(f 1 + f )log 6 (f 1 + f ) ( ) ) 1 +(g 1 + g )log 36 (g 1 + g ), (, ( ) ( ) 1 1 (9 4p) 6(f 1 f )log 36 6 (f 1 f ) ) ) ( 1 +(g 1 g )log 36 (g 1 g ) DC = 1 36p(α β ) ArcTanh [ p(β α ) ] + 7(1 p)log(1 p) + 9(1 + 3p)log(1 + 3p) 36 ( ) 1 18log(1 p (α β ) ) 36 (9 4p)log (9 4p) 36 ( ) ( ) (f 1 f )log 6 (f 1 f ) + 6(f 1 + f )log 6 (f 1 + f ) ( ) ( ) ) 1 1 6(g 1 g )log 36 (g 1 g ) 6(g 1 + g )log 36 (g 1 + g ), where f 1 = 3, f = p(α β ), g 1 = 9 + 4p and g = 4p 9 0α β. We plot in figure 3.9 broadcasting fidelity(fb) with p and α. Further Figure 3.10 shows the complementary nature of F B and T F / DC and figure 3.11 depicts the corresponding bounds. 7

84 Figure 3.9 Plotting F B vs p vs α. The cloning operation applied is local cloning. Figure 3.10 a) Plotting F B with T F. b) Plotting F B with DC In each of the cases the cloning operation applied is local cloning. In TABLE 3.1, we give the broadcasting range of the werner-like states in terms p for the different values of the input state parameter α. We also calculate the change that happened in the teleportation fidelity T F and super dense coding capacity DC as a result of broadcasting. Sum of each of these quantities with broadcasting fidelity F B for a given value of α are provided in the table. This is a clear indication that sum of these quantities is constant for a given value of the input state parameter. In TABLE 3., we give the broadcasting range in terms of α for different values of the classical mixing parameter p. Quite similar to the previous table here also we give the values of the sums ( T F + F B) and ( DC + F B) against the various values of the classical randomness parameter. 73

85 Figure 3.11 a) Plotting ( T F + F B) and b) ( DC + F B) against the parameters α and p of the Werner like state. These are obtained using local cloner.. α Broadcasting ( T F + F B) ( DC + F B) Range < p < p < p < p < p Table 3.1 Broadcasting ranges obtained using local cloners for different values of α. p Broadcasting ( T F + F B) ( DC + F B) Range < α < < α < < α < < α < < α < < α < Table 3. Broadcasting ranges obtained with local cloners for different values of p. 74

86 These sums ( T F + F B) and ( DC + F B) are bounded establishing once again the mutually exclusive nature of both these quantities T F (DC) and F B. Broadcasting by Non Local Cloning: Similar to local cloning, we provide the expression for the relevant quantities and the corresponding graphs. F B = 1 ( ) (1 p)(5 3p) + (1 + 3p)(5 + 9p), 80 T F i = 1 (3 + p + 4pαβ), 6 T F c = 1 (5 + p + 4pαβ), 10 T F = 1 (p + 4pαβ), 15 ( DC i = 1 4 4p(α β ) ArcTanh [ p(β α ) ] + 3(1 p)log(1 p) +(1 + 3p)log(1 + 3p) log ( 1 p (β α ) )), ( DC c = 1 (15 9p)log(5 3p) + (5 + 9p)log(5 + 9p) 0 ) (5 + 3p(β α )log(5 + 3p(β α ) ((5 3p(β α ))log(5 3p(β α )), DC = 1 0 ( 0p(α β ) ArcTanh [ p(β α ) ] + 15(1 p)log(1 p) + 5(1 + 3p)log(1 + 3p) log ( 1 p (β α ) ) 3(5 3p)log(5 3p) (5 + 9p)log(5 + 9p) ) +(5 + 3p(β α )log(5 + 3p(β α ) + ((5 3p(β α ))log(5 3p(β α )). Quite similar to the local cloning situation here also we provide two different tables for detailed analysis of the broadcasting range. In TABLE 3.3, we give the broadcasting range in terms of the classical mixing parameter p for given values of input state parameter α. In this table we have also given the values of ( T F + F B) and ( DC + F B) for the same set of values of the input state parameter. 75

87 Figure 3.1 Plotting F B vs p vs α. The cloning operation applied is non-local cloning. Figure 3.13 a) Plotting F B with T F. b) Plotting F B with DC In each of the cases the cloning operation applied is non-local cloning. Figure 3.14 a) Plotting ( T F + F B) and b) ( DC + F B) against the parameters α and p of the Werner like state. These are obtained using non-local cloner.. In TABLE 3.4, we give the range of broadcasting in terms of the input state parameter α for given values of classical mixing parameter p and for same values of p we have also provided the values of the sums ( T F + F B) and ( DC + F B). 76

88 α Broadcasting ( T F + F B) ( DC + F B) Range < p < p < p < p < p Table 3.3 The broadcasting ranges obtained with a nonlocal cloner for different values of the input state parameter (α ). p Broadcasting ( T F + F B) ( DC + F B) Range < α < < α < < α < < α < < α < < α < Table 3.4 The broadcasting ranges obtained with a nonlocal cloner for different values of the classical mixing parameter (p). In case of non local cloning we also did the similar plot where the figure [3.14] shows that each of these quantities ( T F + F B) and ( DC + F B) are bounded against the parameters α and p. Next we consider the example of Bell diagonal states and the broadcasting of entanglement via local copying with these entangled states as resource. Example C: Bell Diagonal States The bell diagonal state is given by ρ 1 = 1 4 (I I + c 1σ 1 σ 1 + c σ σ + c 3 σ 3 σ 3 ), (3.5) where σ 1, σ, σ 3 are pauli matrices and 1 c 1 c, 1 c 1 and 1 c 3 1. Broadcasting by Local Cloning: 77

89 First, we give the analytical expressions for the relevant quantities like broadcasting fidelity, teleportation fidelity and superdense coding capacity. Then, we will compute the values for different intervals of c 1, c and c 3. F B = T F i = 1 6 ( (9 4c1 + 4c + 4c 3 )(1 c 1 + c + c 3 ) + (9 + 4c 1 + 4c 4c 3 )(1 + c 1 + c c 3 ) + (9 + 4c 1 4c + 4c 3 )(1 + c 1 c + c 3 ) + (9 4c 1 4c 4c 3 )(1 c 1 c c 3 )), ) (3 + c 1 + c + c 3 (7 + 4c 1 + 4c + 4c 3 ) T F c = 1 54 ) (c 1 + c + c 3 T F = 5 54 ( DC i = 1 4 DC c = 1 36 DC = 1 36,,, (1 c 1 c c 3 )log(1 c 1 c c 3 ) + (1 + c 1 + c c 3 )log(1 + c 1 + c c 3 ) +(1 + c 1 c + c 3 )log(1 + c 1 c + c 3 ) + (1 c 1 + c + c 3 )log(1 c 1 + c + c 3 ) ( 7log3 + (9 4c 1 4c 4c 3 )log(9 4c 1 4c 4c 3 ) +(9 + 4c 1 + 4c 4c 3 )log(9 + 4c 1 + 4c 4c 3 ) +(9 + 4c 1 4c + 4c 3 )log(9 + 4c 1 4c + 4c 3 ) +(9 4c 1 + 4c + 4c 3 )log(9 4c 1 + 4c + 4c 3 ) ( 9(1 c 1 c c 3 )log(1 c 1 c c 3 ) + 9(1 + c 1 + c c 3 )log(1 + c 1 + c c 3 ) +9(1 + c 1 c + c 3 )log(1 + c 1 c + c 3 ) + 9(1 c 1 + c + c 3 )log(1 c 1 + c + c 3 ) +7log3 (9 4c 1 4c 4c 3 )log(9 4c 1 4c 4c 3 ) (9 + 4c 1 + 4c 4c 3 )log(9 + 4c 1 + 4c 4c 3 ) (9 + 4c 1 4c + 4c 3 )log(9 + 4c 1 4c + 4c 3 ) ) (9 4c 1 + 4c + 4c 3 )log(9 4c 1 + 4c + 4c 3 ) ),. ), In the TABLE 3.5, we give the broadcasting range of Bell-diagonal states ρ 1 for different values of the first two input state parameters c 1, c. The range is given over the third input state parameter c 3, between the valid zone from 1 to 5 8 or 5 8 to 1. 78

90 c1 c c3 ( T F + F B)max ( DC + F B)max c c3 < c3 < c3 < Table 3.5 Broadcasting ranges along with the maximum values of the sums ( T F + F B)max and ( DC + F B)max obtained with local cloners for different valid values of input state parameters c 1 and c. We have also provided the maximum values of the sums ( T F + F B)max and ( DC + F B)max for this broadcasting range of the input parameter c 3. In this table, we restrict our results only to the negative range of inputs for c 1 and c. The broadcasting range in terms of c 3 will remain unchanged when corresponding positive values of c 1 and c are substituted in Eq. (.41). Broadcasting by Non Local Cloning: Similar to the previous cases, here also we give the mathematical expressions for all the quantities. F B = 1 80( (5 3c1 + 3c + 3c 3 )(1 c 1 + c + c 3 ) + (5 + 3c 1 3c + 3c 3 )(1 + c 1 c + c 3 ) + (5 + 3c 1 + 3c 3c 3 )(1 + c 1 + c c 3 ) + (5 3c 1 3c 3c 3 )(1 c 1 c c 3 )), ) T F i = (3 1 + c 1 + c + c 3, 6 ) T F c = (5 1 + c 1 + c + c 3, 10 ) T F = (c c + c 3, 15 ( DC i = 1 (1 c 1 + c + c 3 )log(1 c 1 + c + c 3 ) + (1 + c 1 c + c 3 )log(1 + c 1 c + c 3 ) 4 ) +(1 + c 1 + c c 3 )log(1 + c 1 + c c 3 ) + (1 c 1 c c 3 )log(1 c 1 c c 3 ), 79

91 DC c = 1 0 ( 0log5 + (5 3c 1 + 3c + 3c 3 )log(5 3c 1 + 3c + 3c 3 ) +(5 + 3c 1 3c + 3c 3 )log(5 + 3c 1 3c + 3c 3 ) DC = 1 0 +(5 + 3c 1 + 3c 3c 3 )log(5 + 3c 1 + 3c 3c 3 ) +(5 3c 1 3c 3c 3 )log(5 3c 1 3c 3c 3 ) ( 5(1 c 1 + c + c 3 )log(1 c 1 + c + c 3 ) + 5(1 + c 1 c + c 3 )log(1 + c 1 c + c 3 ) +5(1 + c 1 + c c 3 )log(1 + c 1 + c c 3 ) + 5(1 c 1 c c 3 )log(1 c 1 c c 3 ) +0log5 (5 3c 1 + 3c + 3c 3 )log(5 3c 1 + 3c + 3c 3 ) (5 + 3c 1 3c + 3c 3 )log(5 + 3c 1 3c + 3c 3 ) (5 + 3c 1 + 3c 3c 3 )log(5 + 3c 1 + 3c 3c 3 ) ) (5 3c 1 3c 3c 3 )log(5 3c 1 3c 3c 3 ) Hence in TABLE 3.6, we give the broadcasting range of bell-diagonal states ρ 1 for different values of the first two input state parameters c 1, c. The range is given in terms of the third input state c 3, between the valid zone from 1 to 1 3 or 1 3 to 1. In addition to that we have also given the maximum values of the sums ( T F + F B)max and ( DC + F B)max in that broadcastable zone to show the complementary nature of these quantities. In this table, we restrict our results only to the negative range of inputs for c 1 and c. The broadcasting range in terms of c 3 remains unchanged when corresponding positive values of c 1 and c are substituted in equation.56. c1 c c3 ( T F + F B)max ( DC + F B)max c c3 < c3 < c3 < Table 3.6 Broadcasting ranges along with the maximum values of the sums ( T F + F B)max and ( DC + F B)max obtained with non local cloners for different valid values of input state parameters c 1 and c. ), Complementarity in Broadcasting multiple copies In this section we extend our result to a situation where we have used 1 M state independent local as well as non local cloning machine on our initial resource state ρ 1 to broadcast more than two copies 80

92 of entanglement (M copies). As discussed in chapter the local cloning transformations for 1 M cloning is given by U 1,M 0 R = U 1,M 1 R = M 1 j=0 M 1 j=0 α j (M j)0, j1 R j, (3.6) α M 1 j (M 1 j)0, (j + 1)1 R j, (3.7) where (M j) α j = M(M + 1) and R is the machine state before cloning R i and R j are the machine states after cloning. Similarly, for 1 M non-local cloner is defined as (3.8) U 1,M ψ i Σ (M 1) R = M 1 j=0 α j n (M j) ψ i, j ψ l R jl, (3.9) where ψ 1 = 00, ψ = 01, ψ 3 = 10 and ψ 4 = 11. Σ (M 1) are M 1 blank states. R is the initial machine state. R jl are the machine states after cloning. α j is the probability that there are j errors out of M cloned state. Also, the machine states can be represented as R jl = (M 1 j) ψ i, j ψ l. k=0 (M j) n α j = M 1 (3.10) j=0 (M j) n, where n is the number of ways in which j states can be chosen out of available basis set { ψ 1, ψ, ψ 3, ψ 4 } and i l. Till now, we have seen how the quantities broadcasting fidelity, teleportation fidelity and superdense coding capacity are related to each other in the context of broadcasting of entanglement through cloning the states. The plots show the complementarity nature of the relevant quantities. Thats how, the expected behaviour of better cloning and less loss of information processing capabilities are reinforced. However, it is also interesting to observe that there has to be a trade-off between two quantities broadcasting fidelity(fb) and loss in information processing capability( T F / DC). One cannot go till its maximum value and bringing the other to its minimum value. There has to be a particular limit to these values when considering a range of original states. We give that bound in the form of summation of the two complementary quantities in the next few tables. In the tables NA denotes that the broadcasting is not possible in that range of states. 81

93 No. of copies copies 3 copies 4 copies 5 copies α p Range ( T F + p Range ( T F + p Range ( T F + p Range ( T F + F B)max F B)max F B)max F B)max NA NA Table 3.7 [Werner Like States] The complementarity bound for teleportation fidelity and broadcasting fidelity obtained with a 1 M non local cloner for different values of the input state parameter(α ) where M =, 3, 4, 5 No. of copies copies 3 copies 4 copies 5 copies p α Range ( T F + α Range ( T F + α Range ( T F + α Range ( T F + F B)max F B)max F B)max F B)max NA NA NA NA NA NA N A N A N A Table 3.8 [Werner Like States] The complementarity bound for teleportation fidelity and broadcasting fidelity obtained with a 1 M non local cloner for different values of the input state parameter(p) where M =, 3, 4, 5 8

94 No. of copies copies 3 copies 4 copies 5 copies α p Range ( DC + F B)max p Range ( DC + F B)max p Range ( DC + F B)max p Range ( DC + F B)max NA NA Table 3.9 [Werner Like States] The complementarity bound for super dense coding capacity and broadcasting fidelity obtained with a 1 M non local cloner for different values of the input state parameter(α ) where M =, 3, 4, 5 No. of copies copies 3 copies 4 copies 5 copies p α Range ( DC + F B)max α Range ( DC + F B)max α Range ( DC + F B)max α Range ( DC + F B)max NA NA NA NA NA NA N A N A N A Table 3.10 [Werner Like States] The complementarity bound for super dense coding capacity and broadcasting fidelity obtained with a 1 M local cloner for different values of the input state parameter(p) where M =, 3, 4, 5 83

95 No. of copies copies 3 copies 4 copies 5 copies c3 c c1 ( T F + c1 ( T F + c1 ( T F + c1 ( T F + Range F B)max Range F B)max Range F B)max Range F B)max NA NA NA N A N A N A Table 3.11 [Bell Diagonal States] The complementarity bound for teleportation fidelity and broadcasting fidelity obtained with a 1 M local cloner for different values of the input state parameter(c, c 3 ) where M =, 3, 4, 5 No. of copies copies 3 copies 4 copies 5 copies c3 c c1 Range ( DC + F B)max c1 Range ( DC + F B)max c1 Range ( DC + F B)max c1 Range ( DC + F B)max N A N A N A N A N A N A N A NA 1 NA 1 NA 1 NA Table 3.1 [Bell Diagonal States] The complementarity bound for super dense coding capacity and broadcasting fidelity obtained with a 1 M non local cloner for different values of the input state parameter(c, c 3 ) where M =, 3, 4, 5 84

96 Chapter 4 Conclusions We conclude this thesis by summarizing the work done and the results obtained. For a dimensional system, we used Buzek-Hillery cloners for broadcasting entanglement both locally and non-locally. We extended the study by making more than copies, for that we used Gisin-Massar cloner. We also gave the corresponding transformations for non-local cloner. We found the broadcasting range for different types of states such as Werner like and Bell diagonal. We observed that the broadcasting range is broader in case of local cloning as compared to non-local cloning. Also, we can broadcast only copies in case of local cloning. However, it can be broadcast till 5 copies in case of non-local cloning. In that case, the broadcasting range becomes shorter as we move from cloning copies to 5 copies. These results are explainable as in local cloning there are more number of machine states as compared to the process of non-local cloning. So, the entanglement is distributed widely in local cloning, resultantly transferring less to the output states. In the next part of the work we discuss the complementarity involving broadcasting fidelity and the information processing tasks such as quantum teleportation and superdense coding. We exemplify the relationship between broadcasting fidelity and the change in telaportation fidelity after broadcasting by plotting the same for states such as non-maximally entangled states and werner like states. The complementarity is further depicted by plotting the summation of these two quantities with respect to the corresponding states. We also plot the same for the most generalized -qubits states. We observe that the sum is always bounded by a number less than in case of teleportation fidelity and less than 3 in case of superdense coding capacity. We tabulate the bounds for werner-like states and bell-diagonal states for different ranges. We found that in case of Bell-diagonal states, we are able to clone only till 3 copies. In this thesis we have considered broadcasting fidelity to analyse the effect of cloning on the information processing tasks. However, this is also important to think that along with cloning the states, entanglement is also being cloned. And the information processing tasks are dependent on how much entanglement is transferred in this cloning process. Concurrence is a widely accepted measure to quantify the presence of entanglement in a system. So, we can also analyse all these results using difference 85

97 of concurrence between original and cloned states instead of broadcasting fidelity. It will be interesting to see how the difference between the values of original and cloned states for the quantities concurrence and the information processing capabilities like telporation fidelity and superdense coding capacities vary with each other. In future, it is also interesting to use the less entangled cloned states to make a more entangled state and observe the complementarity between the dual processes of compression(distillation) and decompression(broadcasting). The expected result should demonstrate the trade-off between the fidelities of both the processes. 86

98 Related Publications Accepted Poster Complementarity in Quantum Information Processing Tasks Jaya Chaubey*, Sourav Chatterjee, Aditya Jain and Indranil Chakrabarty. TQC017 - June 017, Theory of Quantum Computation, Communication and Cryptography, Paris - Universit Pierre et Marie Curie. Research Article Complementarity in Quantum Information Processing Tasks Jaya Chaubey*, Sourav Chatterjee, and Indranil Chakrabarty. arxiv: (016), (Submitted, Journal work in progress). 87

99 Bibliography [1] Bell-diagonal state quantiki. [Online; accessed 15-June-016]. [] No-broadcast theorem Wikipedia, the free encyclopedia. [Online; accessed 15-June-016]. [3] Proceedings of IEEE International Conference on Computers, System and Signal Processing, Bangalore, India, [4] The most beautiful experiment. Physics World, 00. [5] B. P. A. Einstein and N. Rosen. Phys. Rev., 47(1):777, [6] S. Adhikari, I. Chakrabarty, and P. Agrawal. Quant. Inf. Proc., 1:053, 01. [7] S. Adhikari, I. Chakrabarty, and B. S. Choudhary. J. Phys. A, 39:8439, 006. [8] S. Adhikari and B. S. Choudhury. Phys. Rev. A, 74:0333, 006. [9] S. Adhikari, N. Ganguly, I. Chakrabarty, and B. S. Choudhury. J. Phys. A: Math. Theor., 41:41530, 008. [10] S. Adhikari, A. K. Pati, I. Chakrabarty, and B. S. Choudhury. Quantum. Inf. Process., 6:197, 007. [11] P. Agrawal and A. K. Pati. Phys. Lett. A, 305:1, 00. [1] K. Azuma, J. Shimamura, M. Koashi, and N. Imoto. Phys. Rev. A., 7:03335, 005. [13] C. B. B. Dakic and V. Vedral. Phys. Rev. Lett., 105:19050, 010. [14] S. Bandyopadhyay and G. Kar. Phys. Rev. A, 60:396, [15] H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, and B. Schumacher. Phys. Rev. Lett., 76:818, [16] B. Bellomo, R. L. Franco, and G. Compagno. Phys. Rev. A., 77:0334, 008. [17] B. Bellomo, R. L. Franco, and G. Compagno. Phys. Rev. A., 77:0334, 008. [18] C. H. Bennett, H. J. Bernsteina, S. Popescu, and B. Schumacher. Phys. Rev., 53:046, [19] N. Bohr. Nature, 11:50, 198. [0] S. Bose, V. Vedral, and P. L. Knight. Phys. Rev. A., 57:8, [1] D. BruB, P. DiVincenzo, A. Ekert, C. A. Fuchs, C. Macchiavello, and J. A. Smolin. Phys. Rev. A, 57:368, [] D. Bruss, G. M. DAriano, M. Lewenstein, C. Macchi-avello, A. Sen(De), and U. Sen. Phys. Rev. Lett., 93:10501, 004. [3] D. Bruss, G. M. DAriano, M. Lewenstein, C. Macchiavello, A. Sen(De), and U. Sen. Int. J. Quant. Inf., 4:415, 006. [4] D. Bures. Trans. Am. Math. Soc., 135:199,

100 [5] V. Buzek and M. Hillery. Phys. Rev. A, 54:1844, [6] V. Buzek and M. Hillery. Phys. Rev. Lett., 81:5003, [7] V. Buzek, V. Vedral, M. B. Plenio, P. L. Knight, and M. Hillery. Phys. Rev. A, 55:337, [8] H.-S. S. C. Li and L. Zhou. Journal of Optics B: Quantum semiclass. opt., 5:155, 003. [9] A. Cabello. Phys. Rev. A., 61:0531, 000. [30] D. Cavalcanti, A. Acn, N. Brunner, and T. Vrtesi. Phys. Rev. A, 87:04104, 003. [31] N. J. Cerf. J. Mod. Opt., 47:187, 000. [3] N. J. Cerf. Phys. Rev. Lett., 84:4497, 000. [33] I. Chakrabarty. Int. J. Quant. Inf., 7:559, 009. [34] I. Chakrabarty. Eur. Phys. J. D, 57:65 69, 010. [35] I. Chakrabarty, P. Agrawal, and A. K. Pati. Eur. Phys.J D, 65:605, 011. [36] I. Chakrabarty, P. Agrawal, and A. K. Pati. Quantum Information and Computation, 1:071, 01. [37] I. Chakrabarty, S. Banerjee, and N. Siddharth. Quant. Inf. Comp., 11:0541, 011. [38] I. Chakrabarty and B. S. Choudhary. arxiv: [39] S. Chatterjee, S. Sazim, and I. Chakrabarty. Phys. Rev. A, 93:04309, 016. [40] C.H.Bennett and S.J.Weisner. Phys. Rev. Lett., 69:881, 199. [41] A. Chefles and S. M. Barnett. Phys. Rev. A., 60:136, [4] D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera. Phys. Rev. Lett., 77:818, [43] P. A. M. Dirac. The principles of quantum mechanics. Clarendon Press, [44] L. M. Duan and G. C. Guo. Phys. Rev. Lett., 80:4999, [45] A. Einstein, B. Podolsky, and N. Rosen. Phys. Rev., 47:777, [46] A. Ekert. Phys. Rev. Lett., 67:661, [47] H. Fan, K.Matsumoto, and M.Wadati. Phys. Rev. A., 64:064301, 001. [48] A. Fedrizzi, B. Skerlak, T. Paterek, M. P. de Almeida, and A. G. White. New Journal of Physics, 13:053038, 011. [49] J. Fiurasek, S. Iblisdir, S. Massar,, and N. J. Cerf. Phys. Rev. A., 65:04030, 00. [50] W. Gerlach and O. Stern. Zeitschrift fr Physik, pages , 19. [51] I. Ghiu. Phys. Rev. A, 67:0133, 003. [5] D. Girolami and G. Adesso. Phys. Rev. A., 83:05108, 011. [53] N. Gisin. Phys. Lett. A, 4:1, [54] N. Gisin and S. Massar. Phys. Rev. Lett., 79:153, [55] L. Goldenberg and L.Vaidman. Phys. Rev. Lett., 75:139, [56] J. B. H. Barnum, M. Leifer, and A. Wilce. Phys. Rev. Lett., 99:40501, 007. [57] D. Harrison. Complementarity and the copenhagen interpretation of quantum mechanics,

101 [58] C. H.Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters. Phys. Rev. Lett., 70:1895, [59] W. Heisenberg. Zeitschrift fr Physik, 43(3-4):17 198, 197. [60] L. Henderson and V. Vedral. J. Phys. A., 34:6899, 001. [61] M. Hillery, V. Buzek, and A. Berthiaume. Phys. Rev. A., 59:189, [6] T. Hiroshima. J. Phys. A Math. Gen., 34:6907, 001. [63] M. Horodecki, P. Horodecki, and R. Horodecki. Phys. Lett. A., 3:1, [64] R. Horodecki, M. Horodecki, and P. Horodecki. Phys. Lett. A, :1, [65] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki. Rev. Mod. Phys., 81:865, 009. [66] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki. Rev. Mod. Phys., 81:865, 009. [67] H.P.Yuen. Phys. Lett. A., 113:405, [68] M. D. Lang and C. M. Caves. Phys. Rev. Lett., 105:150501, 010. [69] L.Hardy and D. D. Song. Phys. Lett. A, 59:331, [70] S. Luo. J. Phys. A., 77:04303, 008. [71] S. Luo. Lett. Math. Phys., 9:143, 010. [7] S. Luo and S. Fu. Phys. Rev. A., 8:03430, 010. [73] S. Luo, N. Li, and X. Cao. Phys. Rev. A, 79:054305, 009. [74] K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral. Rev. Mod. Phys., 84:1655, 01. [75] K. Modi, T. Paterek, W. Son, V. Vedral, and M. Williamson. Phys. Rev. Lett., 104:080501, 010. [76] M.Ray, S. Chatterjee, and I. Chakrabarty. Eur. Phys. J. D, 70:114, 016. [77] W. J. Munro, D. F. V. James, A. G. White, and P. G. Kwiat. Phys. Rev. A., 64:03030, 001. [78] R. Nepal, R. Prabhu, A. S. De, and U. Sen. Phys. Rev. A., 87:03336, 013. [79] H. Ollivier and W. H. Zurek. Phys. Rev. Lett., 88:017901, 001. [80] J. Oppenheim, K. Horodecki, M. Horodecki, P. Horodecki, and R. Horodecki. Phys. Rev. A, 68:0307, 003. [81] J. Oppenheim, K. Horodecki, M. Horodecki, P. Horodecki, R. Horodecki, A. Sen, U. Sen, and B. Synak- Radtke. Phys. Rev. A, 71:06307, 005. [8] A. Peres. Quantum Theory : Concepts and Methods, volume 73. Kluwer Academic Publishers, [83] A. Peres. Phys. Rev. Lett., 77:1413, [84] M. Piani, P. Horodecki, and R. Horodecki. Phys. Rev. Lett., 100:09050, 008. [85] A. M. S. Adhikari and N. Nayak. Phys. Rev. A., 77:04301, 008. [86] J. J. Sakurai. Modern quantum mechanics, [87] S. Sazim and I. Chakrabarty. Eur. J. Phys., 67(8):174, 013. [88] S. K. Sazim, I. Chakrabarty, A. Datta, and A. K. Pati. Phys. Rev. A, 91:06311, 015. [89] S. K. Sazim, I. Chakrabarty, C. Vanarasa, and K. Srinathan. Quant. Inf. Proc., 14:4651, 015. [90] V. Scarani, S. Iblisdir, N.Gisin, and A. Acin. Rev. Mod. Phys., 77:15,

102 [91] E. Schrodinger. Proceedings of the Cambridge Philosophical Society. [9] Z. Shadman, H. Kampermann, C. Macchiavello, and D. Bruss. New J. Phys., 1:07304, 010. [93] P. W. Shor. SIAM J. Comput, 6:1484, [94] P. W. Shor and J. Preskill. Phys. Rev. Lett., 85:441, 000. [95] V. Vedral, M. Plenio, M. Rippin, and P. L. Knight. Phys. Rev. Lett., 78:75, [96] E. H. Walker. The physics of consciousness, 000. [97] T. Wei, K. Nemoto, P. M. Goldbart, P. G. Kwiat, W. J. Munro, and F. Verstraete. Phys. Rev. A., 67:0110, 003. [98] R. F. Werner. Phys. Rev. A., 40:477, [99] W. K. Wootters and W. H. Zurek. Nature, 99:80, 198. [100] J.-S. Zhang and A.-X. Chen. Quant. Phys. Lett., 1:69, 01. [101] M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert. Phys. Rev. Lett., 71:487,

103 1 1 This image is taken from http : // sciencenews.php 9