Study of Gaussian Channels: Classification of One Mode Bosonic Channels, Kraus Representation and a Result From ESD

Transcription

1 Study of Gaussian Channels: Classification of One Mode Bosonic Channels, Author: Kraus Representation and a Result From ESD Tanmay Singal, Junior Research Fellow, The Institute of Mathematical Sciences, Supervisor: Sibasish Ghosh Taramani, Chennai A thesis submitted to the Board of Studies in Physical Science Discipline In partial Fulfillment of Requirements For the Degree of Master of Science of HOMI BHABA NATIONAL INSTITUTE July 1, 011

2 Bonafide Certificate Certified that this dissertation titled Study of Gaussian Channels:Classification of One Mode Bosonic Channels, Kraus Representation and a Result From ESD is the bonafide work of Mr.Tanmay Singal who carried out the project under my supervision. Sibasish Ghosh, Theoretical Physics, Institute of Mathematical Sciences, Chennai Date:

3 Acknowledgements I would like to thank my advisor Sibasish Ghosh for his guidance through the wonderful journey that has culminated in this thesis. He has been instrumental in helping with the work and in providing advice and motivation. The fruitful discussions have helped me understand the problem and will go a long way in my eduation. I also wish to thank Dibyakrupa Sahu for helping me with latex and Professor R. Simon for helping me with some doubts I had.

4 Abstract This review starts with the basic Hilbert space structure for continuous variable systems, explaining the pre-requisites to be able to define a Gaussian state. After that a preliminary study of Gaussian channels is undertaken. The criteria for a quantum channel being Gaussian is laid out. In accordance with these criteria a classification of one-mode bosonic Gaussian channels with a single-mode environment is given. The Kraus representation for the various channels is then obtained and used to verify some of the pre-existing properties of the various channels (non-classicality breaking, entanglement breaking, extremality of the channel, etc). After this the entanglement sudden death (ESD) of a two-mode Gaussian state under the action of a Gaussian channel is studied. This channel comprises of two mutually exclusive channels, each of which is acting on one of the two-modes of the Gaussian states. The channel action comprises of interaction of the mode with a thermal bath. Both the channels, interacting with the two modes separately are at the same temperature. In the study for ESD, it is discovered that for all non-zero temperatures, all entangled-two-mode Gaussian states undergo ESD at some time or the other. For the zero temperature case S. Goyal and S. Ghosh have proved that ESD won t occur for a set kind of states. We have tried to generalize this result i.e. found another set of two mode Gaussian states which also won t undergo ESD under the channel action. It is desired to know if there are some two-mode Gaussian states which will undergo ESD at zero temperature. We find that there are some states which undergo ESD while interacting with a zero-temperature thermal bath. 3

5 Contents 1 Hilbert Space 5 Gaussian States 10 3 Entanglement 15 4 Gaussian Channels 17 5 Classification of One-Mode Gaussian Channel (canonical forms) 5.1 Kraus Representation: One-Mode Bosonic Gaussian Channels Singular Case:A A A B B B C C1 Attenuator Channel C Amplifier Channel D Semi-group Property of One-Mode Bosonic Gaussian Channels Entanglement Sudden Death Master Equation for a Single Mode System Coherent State Representation Covariance Matrix Evolution ESD of a Two Mode Gaussian State Commutativity of Channel-Action with Local Symplectic Transformations Extending the ESD Result for Different States at Zero Temperature States That Won t Undergo ESD Certain Two-Mode Gaussian States Undergo ESD at T=

6 1 Basic Hilbert Space Structure Consider an infinitely dimensional quantum system (e.g. a one-dimensional simple harmonic oscillator) associated with two operators ˆp and ˆq having the commutation relation given by [ˆq, ˆp] = ι 1. Here, ˆq represents the position operator and ˆp the momentum operator. From these two we can define two more operators given by: â = 1 (ˆq + ιˆp) (1) and â = 1 (ˆq ιˆp) () which will have the following commutation relation: [â, â ] = 1 (3) The number operator is then given by ˆN = ââ. Any quantum state is expressible in terms of a density operator, ρ with the following properties: ˆρ 0, ˆρ = ˆρ, T r[ˆρ] = 1. Hermiticity of ˆρ implies that it has a spectral decomposition: ˆρ = n i=1 p i i i where 1 p i 0, n i=1 p i = 1 and i j = δ i,j Consider a state, ρ of an n-mode quantum mechanical radiation field. Define a column vector as follows: ˆR = ˆq 1 ˆp 1 ˆq ˆp... ˆq ṋ (4) p n 1 We take = 1 for convenience. 5

7 where ˆq i and ˆp i denote the position and momentum operators acting on the ith mode.thus the components of ˆR v.i.z. ˆR1, ˆR, ˆR 3, ˆR 4,..., ˆR n denote respectively ˆq 1, ˆp 1, ˆq, ˆp,..., ˆp n.the Hermitian conjugate of ˆR is given by: ˆR = ( ˆq 1 ˆp 1 ˆq ˆp ˆq n ˆp n ) (5) The mean vector is given by: R = T r(ˆρ ˆq 1 ) T r(ˆρ ˆp 1 ) T r(ˆρ ˆq ) T r(ˆρ ˆp )... T r(ˆρ ˆq n ) T r(ˆρ ˆp n ) = ˆq 1 ˆp 1 ˆq ˆp... ˆq n ˆp n (6) Consider the matrix V of correlations defined by: V ij = T r(ˆρ( ˆR i ˆR i )( ˆR j ˆR j )) (7) One can always displace the mean values of the respective position and momentum operators so as to reduce them to zero via displacement operators for any state. In other words, given a state ˆρ, we replace the observables ˆR j by ˆR j ˆR j ˆR j T r(ˆρ ˆR j ). From (7) we can see that this doesn t change the values taken by the second order moments in anyway. From now on we ll assume this to be true. Further we have: V ij = T r(ˆρ ˆR i ˆRj ) = 1 [T r(ˆρ{ ˆR i, R j } [ ˆR i, ˆR j ])] = γ ij + 1 ιω ij (8) Here, { ˆR i, ˆR j } = ˆR i ˆRj + ˆR j ˆRi is the anti-commutation relation between ˆR i and ˆR j and [ ˆR i, ˆR j ] = ˆR i ˆRj + ˆR j ˆRi is the commutation relation between the two. Given that ˆR has been displaced to 0 we get: γ ij = 1 T r(ˆρ{ ˆR i, ˆR j }) (9) and 6

8 Ω ij = ιt r(ˆρ[ ˆR i, ˆR j ]) (10) Here (γ ij ) is called the covariance matrix. It is a real and symmetric matrix. It basically holds the information regarding the second moments of the quantum state. In the following it s shown that (γ ij ) is ( ) T positive semidefinite too. Let v = v 1 v... v n R n be a vector. Then: v T γv = n So γ is positive semi-definite. v i γ ij v j = n v i v j ˆR i ˆRj = ( n v i ˆRi ) ( n i,j=1 i,j=1 i=1 j=1 v j ˆRj ) 0 Ω is called the commutation matrix and has the form: Ω = (11) The evolution of a closed quantum system is obtained by acting unitary transformation on the state.in the Heisenberg picture this transformation would amount to obtaining a new set of position and momentum operators for the respective modes as respective functions of the older ones. The functions should be such that the commutation matrix,ω (with respect to the new position and momentum operators) would remain invariant. The set of all possible such transformations forms a group called the symplectic group, Sp(n, R). Since we are working with the ˆq i and ˆp i operators of an n-mode Hilbert space the representation of Sp(n, R) would be real i.e. the Sp(n, R) matrices which would effect all possible transformations (within a closed system) would have real entries. The defining representation for a matrix element of Sp(n, R) matrix group is as follows: Let S Sp(n, R). Then SΩS T = Ω The change in the position-momentum operators is given by ˆR = S ˆR. Then the new commutation matrix would look like: Ω ij = ι [ ˆR i, ˆR j] = ι [(S ˆR) i, (S ˆR) j ] = ι n k,l=1 S ik [ ˆR k, ˆR l ] S T lj = ι(sωst ) ij = Ω ij. 7

9 Hence a such a transformation would not change the commutation relations between the new position and momentum operators of the respective new modes. Classical systems allow (ˆq i ) and (ˆp i ) to take any values i.e., there is no constraint that these two quantities need to satisfy to represent a possible configuration of the classical system. In quantum systems, the Heisenberg s uncertainty principle imposes constraints in the following form : ˆq ˆp ( 1 {ˆq, ˆp} ) 1 4 (1) Hence, if the covariance matrix has the following form: A 11 A 1 A A 1 A A 3... γ = A 31 A 31 A (13) where A ij is a block matrix. Then (13) implies Det(A ii ) 1.This basically amounts to the 4 following: q i p i ( 1 {q i, p i } ) 1 4 (14) Williamson s Theorem 3 : Given a n n real, symmetric and positive definite matrix γ, there exists a real symplectic matrix S Sp(n, R) such that the transformation the latter effects upon the former is to diagonalize it in such a manner that every eigen value of the former has a degeneracy of i.e. SγS T = diag(κ 1, κ 1, κ, κ,..., κ n, κ n ) (15) where κ i 0 i = 1,,..., n The covariance matrix defined above are real, positive and symmetric. Hence Williamson s Theorem guarantees that there exists a symplectic transformation through which the former covariance matrix is transformed to a doubly degenerate diagonal matrix given by (15). We will call the doubly degenerate diagonal matrix so obtained the symplectic diagonal matrix of a given covariance matrix. Also, the eigen spectrum {κ 1, κ,..., κ n } of the symplectic diagonal matrix corresponding to a given covariance matrix, are Generalized Uncertainty Relations and Coherent and Squeezed States, D. A. Trifonov, 3 R. Simon, S. Chaturvedi, V. Srinivasan, J. Math. Phys. Vol. 40, 3634 (1999) 8

10 known as the latter s symplectic eigen spectrum. Consider the symplectic diagonal matrix of any covariance matrix: γ d = κ κ κ n 0 (16) 0 κ n For the doubly degenerate symplectic diagonal matrix it is easy to make out that the condition (14) amounts to: κ i 1 (17) for i = 1,,..., n. Hence, in the doubly-degenerate symplectic diagonal covariance matrix, the product of the position and momentum variances should be greater than or equal to the Heisenberg uncertainty limit. Applying a symplectic transformation and the resulting intermingling of various modes shouldn t hurt this condition. The covariance condition (14) can be brought down to a more elegant form 4 : γ + ι Ω 0 (18) where Ω = Ω 1 Ω...Ω n and Ω i is the commutation matrix for the ith mode i.e. Ω i = Any real, positive and symmetric matrix satisfying (16) is a bonafide covariance matrix. Instead of expressing the covariance matrix γ as second order moments of the ˆq and ˆp operators we can use theâ and â operators to express second order moments. Define the column vector ˆR (c) as follows: 4 R. Simon, N. Mukunda, B. Dutta, Phys. Rev. A Vol. 49,1567 (1994) 9

11 ˆR (c) â 1 â 1 â â... â n â n = 1 1 ι ι ι ι ι ι ˆq 1 ˆp 1 ˆq ˆp... ˆq ṋ p n (19) For the n-mode system, the complex covariance matrix γ (c) is hence defined by taking the mean of each matrix element in the following operator-matrix: ˆR (c) ( ˆR (c) ) = â 1 â 1 â â... â n â n ( â 1 â 1 â â...â n â n ) (0) Hence, the complex covariance matrix can be obtained as: γ (c) = ΛγΛ T (1) where Λ is the matrix on the left on the RHS in (19). Gaussian States To get to the definition of Gaussian states, we go through the underlying mathematical structure briefly: A little bit about phase space variables in Quantum Mechanics: In QM, for a single particle endowed 10

12 with a position and momentum operator to describe it, the domain of the phase space is R. One can equivalently describe the system using the annihilation â and creation â operators defined in (1) and () respectively. For the latter mode of description, then the phase space variables comprises of the complex plane. Phase space variables play a different role in quantum mechanics than what they do in classical mechanics. In classical mechanics a point in the phase space assigns a unique value to the various degrees of freedom needed to specify the configuration of the corresponding classical system completely. In comparison in quantum mechanics, no system can have unique values assigned to all the various degrees of freedom due to Heisenberg s uncertainty principle. Thus in QM, phase space variables don t adopt the same role they do in CM. In QM they are used for what is known as the Weyl Correspondence which assigns a bijective function on the phase space of the system to Hermitian operators (see (3). A bit of this correspondence will be explained in the material to come. Any operator acting on the single-mode Hilbert Space introduced, whether bounded or unbounded, can be expanded as a function of a series of monomials made up of the creation and annihilation operators (and identity operator corresponding to all the zero order terms in the expansion). This can be proved as follows: Consider any operator, ˆF acting on the Hilbert Space. Expand it in the Fock-State basis: 5 = Now 0 0 = exp( â â). ˆF = n n ˆF m m n,m=0 n ˆF 1 m (â ) n 0 0 â m n! m! n,m=0 A very important class of unitary operators for continuous variable systems are the displacement operators, ˆD(α). For a single mode system these are defined as follows: ˆD(α) = e αâ α â, α R () These operators are functions of the phase-space variables.also, they possess a completeness property, as in, any operator with a finite Hilbert-Schmidt norm can be expanded as a convolution of these displacement operators. This is in accord with the statement made before that any operator can be expanded as a series comprising of annihilation and creation operators. 5 The Fock state basis is the eigenbasis of the number operator. 11

13 ˆF = f(α) ˆD( α)d α (3) where f(α) = T r[ ˆD(α) ˆF ] and f(α) d α = T r( ˆF ˆF ). We see that when ˆF has a finite Hilbert Schimdt norm 6 then the function f(α) will be square-integrable. Now using the Baker-Campell Hausdorff formula we obtain:, D(α) = e α e αâ e α â = e α e α âe αâ (4) Upon expanding the exponentials in (4) we see that there is a difference in the ordering of the creation and annihilation operators in the various expressions corresponding to the same displacement operator. On the right most of (4), the annihilation operators are set to the left which leaves the creation operators on the right. This kind of ordering is known as anti-normal ordering. In the centre of (4) we see that the creation operators are always on the left and hence the annihilation operators are always on the right. Expanding the expression on the RHS of () we see that the annihilation and creation operators are ordered symmetrically i.e. in the expansion, for any monomial of â and â of a particular ordering there is another monomial in which the creation and annihilation operators are swapped, rendering the entire expansion symmetrical with respect to ordering. This kind of ordering in operators is known as symmetrical ordering (or Weyl ordering). Since the creation and annihilation operators don t commute (see (3), to compensate for the different ordering schemes adopted, the expression e ( ± α ) appears as a coefficient for the normal and anti-normal ordering. The ordering of the annihilation and creation operators is generalized as follows: Define ordered displacement operators as: D(α, s) = e s α ˆD(α) (5) The parameter s is called the order parameter. When s = 1, we get the normally ordered displacement operator: D(α, 1) = e α D(α) = e αâ e α â (6) As in, all the monomials appearing in the expansion of D(α, 1) are normally ordered. 6 The Hilbert Schmidt norm for a bounded operator is defined as q T r(â Â). 1

14 When s = -1, we get the anti-normally ordered displacement operators: (α, 1) = e α D(α) = e αâ e α â (7) where all the monomials in the expansion are anti-normally ordered. When s = 0, we get the usual displacement operator in the sense it was defined in (). The corresponding monomials are symmetrically ordered.one can saturate all different kinds of ordering by varying the order-parameter s within and upon the unit circle in the complex plane 7. Now, ˆF = T r[ ˆF ˆD(α)] ˆD(α)d α = T r[ ˆF ˆD(α, s)] ˆD(α, s)d α (8) In (8) we obtain an example of what is known as the Weyl Correspondence. Here an operator ˆF is expanded as a convolution of the s-ordered displacement operator, ˆD(α). Let f(α, s) T r( ˆF ˆD(α, s)) = T r( ˆF ˆD(α)e s α ). Hence we have an operator to function mapping as follows: ˆF f(α, s) (9) When s is varied below zero, due to the presence of the term e ( s α ) in f(α, s), the function f(α, s) may not remain square integrable. This is the reason why not all operators of finite Hilbert-Schimdt norm are expandable as a convolution of anti-normally ordered displacement operator. This can easily be generalized to the multi-mode case where the corresponding displacement operators are merely tensor products of the displacement operators of individual modes. All density operators defined on the Hilbert Space have a finite Hilbert Schmidt norm. Hence these are expandable as the convolution integral as done in (8). When ˆF is a density operator, then the convoluting function is called the state s characteristic function, χ(α). This function is called so because it generates moments for the various monomials of the creation and annihilation operators. The usage of different ordering parameter gives one the moments of correspondingly ordered bosonic operators. The s-ordered-characteristic function is given as: χ(α, s) = T r(ˆρ ˆD(α, s)) (30) Now, differentiating the displacement operators by the phase-space variables, α and α and setting the 7 See Ordered Expansions in Boson Amplitude Operators by K. Cahill and R Glauber, Physical Review Vol. 177,

15 latter to 0, {(â ) n â m } s n+m ˆD(α, s) α n (α ) m α =0 (31) Thus we get, n+m χ(α, s) α n (α ) m α =0 =T r(ˆρ{ n+m ˆD(α, s) α n (α ) m α =0}) = T r(ˆρ{(â ) n â m } s ) = {(â ) n â m } s Instead of working with characteristic functions of varying order, we can work with their fourier transforms as well. The fourier transforms of normally ordered characteristic function is called the Sudarshan Glauber function and is denoted by P (ξ) where ξ denotes the phase space variables as usual. P (ξ) = d αe ξα ξ α χ(α, 1) (3) Glauber stated a criterion which says that a system is considered classical if the Sudarshan Glauber function, P (α) is non-negative for every point α on the phase space. Equivalent to this is the statement that the covariance matrix defined in (9) satisfies γ 1I n 0. 8 The fourier transform of the symmetrically ordered characteristic function is the called the Wigner Distribution function, W (ξ). W (ξ) = d αe ξα ξ α χ(α, 0) (33) The fourier transform of the anti-normally ordered characteristic function is the called the Husimi function, Q(ξ). Q(ξ) = d αe ξα ξ α χ(α, 1) (34) After havin defined the necessary pre-requisites we now move on to what a Gaussian state is. Gaussian states are those quantum states whose symmetrically ordered characteristic function are Gaus- 8 See equation (18) in Inseparability Criterion for Continuous Variable Systems by Lu-Ming Duan, H. Giedke, J. I. Cirac and Zoller, Phys Rev. Lett. Vol. 84, 7. 14

16 sian functions in the phase space variables. χ(α, 0) = exp( 1 α γ (c) α + ι ˆR (c) α) (35) These are an important class of states, commonly encountered in nature and easy to produce in the laboratory. 3 Entanglement Criteria of Two Mode Gaussian States Consider a two mode Gaussian state. A Gaussian state is completely defined by its covariance matrix γ and the mean vector ˆR. Let one of the modes be denoted by 1 and the other by. The necessary and sufficient condition for γ to be a bonafide covariance matrix is given by (18). Consider any bonafide covariance matrix, γ of a two-mode system. This matrix can be brought into a canonical form using local symplectic transformations on both the modes - 1 and in the following way: γ 0 = (S 1 S )γ(s 1 T S T ) (36) where γ 0 is the following form: γ 0 = a 0 c 0 0 a 0 d c 0 b 0 0 d 0 b (37) This can be done in the following way. Let: γ = A C T C B (38) A(B) submatrix is the covariance matrix of subsystem 1(). Hence using Williamson s theorem one can diagonalize this matrix using S 1() to turn it into a multiple of I. Let s suppose the cross-covariance matrix C changes from C to C. The cross-covariance matrix C contains information about the correlations between the two systems. Consider the singular value decomposition of C : C = O 1 C 0 O (39) 15

17 where C 0 is a diagonal matrix with and O 1() SO(). Upon choosing the following: S 1() = S 1()O 1 1() (40) we get the covariance matrix in the canonical form (36) where a 1, b 1 (see (17). The necessary and sufficient condition for 4 4 matrix of the form of (36) to be a covariance matrix is then given by: 9 : 4(ab c )(ab d ) a + b + cd 1 4 (41) To obtain the conditions above, one needs to find the eigen values of the matrix γ 0 + ι Ω and apply the condition that the obtained eigenvalues be non-negative (41) has been obtained for a canonical form of the covariance matrix (36). Any 4 4 covariance matrix can be brought to this canonical form by the method described above. Upon expanding the terms in (41) one obtains: a b + ( 1 4 cd) ab(c + d ) 1 4 (a + b ) 0 (4) where we see that:a = DetA,b = DetB,cd = DetC and ab(c + d ) = T r(aωcωbωc T Ω) when the matrices A,B and C are given by (36) All these quantities (DetA,DetB,DetC and T r(aωcωbωc T Ω)) are invariant under local symplectic transformations i.e. invariant under any transformation S Sp(, R) Sp(, R). Now, the eigen values of a matrix don t change sign under an Sp(, R) Sp(, R) transformation 10 Thus γ 0 0 γ 0. Thus (41) (or (4)) can also be written as: detadetb + ( 1 4 detc) T r(ajcjbjc T J) 1 (deta + detb) (43) 4 This gives the necessary and sufficient criteria for any 4 4 matrix to be a bonafide covariance matrix for some two-mode state. Peres gave a necessary criteria for the separability of bi-partite density matrices 11. The necessary criteria states that if the partial transpose of the density matrix with respect to any subsystem is not a bonafide density matrix, then the system (represented by the density matrix) is necessarily entangled. Hence 9 Peres-Horodecki Separability Criterion for Continuous Variable Systems R. Simon, Phys Rev. Lett. Vol. 84, Let A be a positive semi-definite n n matrix i.e. v T Av 0, v R n. Hence v T (SAS T )v 0 v R n. 11 Separability Criteria for Density Matrices A. Peres, Phys. Rev. Lett. Vol. 77,

18 positivity of the partial transpose is a necessary condition for the separability of the state. For the case of -mode gaussian states this was shown to be a sufficient condition by Simon(see footnote (9)). There is a physical interpretation to partial transposition in the case of continuous systems. That is that density-matrix transposition implies time reversal i.e. transposition implies ˆp ˆp. Thus in case of partial transposition, the momentum operator of the sub-system being transposed will change signature. In our case let system- be transposed. Then resulting change covariance matrix γ (38) is given by: γ diag(1, 1, 1, 1)γdiag(1, 1, 1, 1) (44) The only term that undergoes any change in (44) is DetC which undergoes a change of signature. Hence the condition of separability of the covariance matrix is as follows: detadetb + ( 1 4 detc ) T r(ajcjbjc T J) 1 (deta + detb) (45) 4 We can see from (45) in case DetC 0, the state is necessarily separable. For a two-mode Gaussian state to be entangled (43) would have to be satisfied but not (45). 4 Gaussian Channels A quantum channel Φ is a completely positive trace preserving map from the set of operators on one Hilbert space to the set of operators on another Hilbert space i.e. Φ: B(H A ) B(H B ) (where B(H) denotes the set of all bounded operators acting on the Hilbert Space H and H A and H B denote two Hilbert Space) is a quantum channel if 1 : 1. Trace Preserving Condition:- T r( ˆF A ) = T r(φ( ˆF A ))for all ˆ FA B (46). Positivity Condition:- Fˆ A 0 Φ( F ˆ A ) 0 (47) 3. Completely Positivity Preserving Condition:- If H C denotes a third Hilbert space of, say, dimension n 1 Quantum Information and Quantum Computation, Nielsen and Chuang, 8..4 Axiomatic Approach to Quantum Operations, page

19 and ˆF B(H A H C ) with ˆF 0. Then: (Φ I n )( ˆF ) 0 (48) positive integer n and I n : B(H C ) B(H C ) is the identity map. A channel can be given a mathematical form by means of what is called a Kraus Represenation which is based on the interaction between the system (called Principal System) and a modelling environment. Since it s the principal system we are interested in there is a certain amount of flexibility with regards to the environment used to obtain the Kraus representation for a channel. A Kraus Representation is a set of operators acting on the input operator to give the output operators. This can be depicted as follows: Let ˆρ s be in initial state of the principal system. We assume an environment coupled to the principal system and that the state ˆρ in of the entire system is initially in a product state with some initial pure state of the environment, 1 E 1 : ˆρ in = ˆρ s 1 E 1 (49) Let the joint initial state undergo a unitary evolution, Û under a Hamiltonian defined on the joint Hilbert space of the principal system and the environment to reach a final state, ˆρ out. ˆρ out = Û ˆρ inû (50) Consider an ordered orthonormal basis for the environment system featuring 1 E as a basis-vector, { i E }. Tracing out the environment system with respect to this basis we get the desired output state for the channel ˆρ s: ˆρ s = n i Û 1 ˆρ s 1 Û i = i=1 n  i ˆρ s  i (51) i=1 where the set of operators {Âi} are called the Kraus Operators and these operators satisfy the following trace-preserving condition: n  iâi = I n, the n n identity matrix. (5) i=1 Given that we have found a set of Kraus Operators which satisfy (5), the map induced by the set of Kraus Operators is completely positive and trace-preserving. 18

20 Let our principle system comprise of n s modes and the environment comprise of n E modes.consider S Sp((n s + n E ), R). Let it act on the position and momentum vector of the system environment. ˆR = S ˆR (53) where ˆR = ˆR s ˆR E and ˆR is given by (4) for the principal system and the environment. Sp((n s + n E ), R) possess a unitary representation 13 which acts on the tensor product Hilbert Space of the principal system and environment. The unitary operators of this representation are generated by quadratic Hermitian operators. Let Û be the unitary operator acting on the Hilbert, H S H E space of the entire system which corresponds to the symplectic transformation S in its defining representation matrix form. Equation (53) can also be written as: ˆR i = Û ˆR i Û (54) where ˆR i is the ith component of the position-momentum vector ˆR. From (54), we can see that the symplectic transformation, S on the joint system can considered equivalent to performing a unitary transformation, U on the Hilbert space H S H E joint system. We are limiting our discussion to the unitary transformations generated by quadratic hamiltonians. One can see that due to (50) the covariance matrix, γ = γ s γ E of the system+environment will undergo the following transformation: γ = SγS T (55) Hence through this transformation γ s γ s where γ s represents the new covariance matrix of the principal system. It can be easily checked that γ s satisfies the conditions for covariance (18) if γ s does. Consider the two mode case - with one mode principal system and one mode environment. In the following we work in the Heisenberg picture and real variables: Let the symmetrically ordered characteristic function of the joint state (i.e. one mode principal system and one-mode environment) of the system be χ(z, 0) = χ W (z). Here z R 4 represents the real phase-space variables. The phase space variables for the principal system will be denoted by z s = (x s, y s ) and those for the environment will be denoted as z E = (x E, y E ). Hence we get that: 13 Real symplectic groups in quantum mechanics and optics, A. Dutta, N. Mukunda and R. Simon, Pramana Journal of Physics, Vol. 45, pp

21 z = z s z E (56) χ W (z s z E ) = T r[(ˆρ s 1 E 1 ) ˆD(z s z E )] (57) After (54) is performed one again evaluates the characteristic function: χ W (z s z E ) = T r[(ρ s 1 E 1 ) exp(ι ˆR T S T z)] (58) We divide the matrix of the symplectic transformation S into blocks as follows: S = X Y X Y (59) Hence (58) becomes: χ W (z s z E ) = T r(ρ s 1 E 1 exp(ι(x ˆR s + Y ˆ R E ) T z s + ι(x ˆRs + Y ˆRE ) T z E )) (60) We aren t interested in the characteristic function of the entire system but only the principal system so we set the phase space variables z E = x y = 0 0. Hence the characteristic function of the principal system is now given by: χ s W (z s ) = T r(ρ s 1 E 1 exp(ι(x ˆR s + Y ˆ R E ) T z s )) = χ s W (X T z)χ E W (Y T z s ) (61) Hence the final characteristic function is a product of the initial system characteristic function and the environment state characteristic function. We are interested in a class of channels called Gaussian Channels wherein Gaussian states are tranformed ONLY into Gaussian states by the channel. From (61) it s understood that this is possible only if the environment used is also in a Gaussian state. Hence the conditions for a channel being a Gaussian channel have been obtained as : 1. The unitary interaction between the system and environment should have a quadratic generator corresponding to some symplectic transformation.. The environment should also have to be in the Gaussian state initially. 0

22 The displacement operator of the principal system, ˆD(zs ) undergoes a transformation as follows: ˆD(z s ) ˆD(X T z s )χ E W (Y T z s ) = ˆD(X T z s )f(z) (6) The function f(z s ) is the extra noise that is added as a result of the interaction. Since the environment has a Gaussian characteristic function, f(z s ) is Gaussian in nature. No arbitrary function can act as extra noise because a certain set of completely-positivity conditions need to be satisfied for this. These completely-positivity conditions are as follows 14 : form. Since f(z) 15 is gaussian in nature, it will be of the form f(z) = exp( 1 α(z, z)) where α is in a quadratic If α E is the covariance matrix for the environment then α(z 1, z ) z T 1 Y T α E Y z (63) For any finite set of z belonging to the phase-space of the principal system construct the matrix M as follows: M ij α(z i, z j ) ι (z i, z j ) + ι (XT z i, X T z j ) (64) where (z i, z j ) = x j y i x i y j is the symplectic product of two phase space vectors. The positivity of the above matrix for any set of phase space vectors, z = x y guarantees the complete positivity of the Gaussian Channel. This test can be simplified to quite a degree. Let s construct the matrix M taking n phase-space vectors {z i } i=1. We get: M ij = z T i (α ι (detx 1)Ω)z j = z T i βz j (65) where β α ι (detx 1)Ω16. Let η be an n-dimensional complex vector. If M is to be positive then η Mη 0. Testing this: n n η Mη = ( ηi z i )β( η j z j ) (66) i=1 14 A.S. Holevo,One-mode quantum Gaussian Channels(006) quant-ph/ For the sake of convenience from now on, unless otherwise specified, z s will be written as z to represent the phase space variables for the principal system. 16 α = Y T α E Y where α E is the initial covariance matrix of the environment system j=1 1

23 Now, n i=1 η iz i can be considered an arbitrary vector of the phase space. Hence we can see that iff β 0 then M 0. 5 Classification of One-Mode Gaussian Channel (canonical forms) We study the classification of one-mode Bosonic Gaussian Channels which result from the interaction with a single mode environment. This classification is based on properties (i.e. the determinant, rank) of the sub-matrices X, Y, X and Y of the symplectic matrix (see (59) which acts on the joint system of principal system and environment. 17. We are interested to obtain only the canonical forms of these Gaussian Channels and work towards that end. The conditions imposed upon X, Y, X and Y so that they be sub-matrices of a symplectic matrix S Sp(4, R): X Y Ω 0 XT X T X Y 0 Ω Y T Y T = XΩXT + Y ΩY T XΩX T + Y ΩY T X ΩX T + Y ΩY T X ΩX T + Y ΩY T = Ω 0 0 Ω (67) (5) imposes 3 conditions on the sub-matrices X, Y, X and Y. XΩX T + Y ΩY T = Ω (68) X ΩX T + Y ΩY T = Ω (69) XΩX T + Y ΩY T = 0 (70) For any matrix (not necessarily symplectic) P : P ΩP T = 0 detp detp 0 (71) From (71) we can re-write the condition (68) as: 17 F Cariso, V. Giovannetti, A. S. Holevo, New Journal of Physics 8 (310) (006) detx + dety = 1 (7)

24 A similar deduction can be made for X and Y matrices too. We briefly summarize the channel-action for this case, additionally explaining how one obtains the canonical forms (or simplest forms) for a type of one-mode Gaussian channel 18. We are given a one-mode Gaussian state. This can be ANY one-mode Gaussian state and there are no restrictions on this. Before the channel is applied the position and momentum operators of the principal system are ˆq s and ˆp s respectively and that for the environment are ˆq E and ˆp E. The symmetrically ordered displacement operators of the joint system are given by ˆD(z) = exp(ι ˆR T z) where ˆR and z are given by (4) and (56). The characteristic funtion corresponding to the joint system ˆρ in = ˆρ s ρˆ E is hence given (30) whilst keeping the order parameter, s = 0. Since the joint initial state (i.e. before any channel action) is a product state of the principal system and the environment, the characteristic function of the joint initial system is the product of the respective individual characteristic functions of the principal system and the environment i.e. χ in W (z) = χ s W (z s )χ E W (z E ) (73) If α s and α E are the respective covariance matrices of the principal system and the environment before the channel action, then the characteristic function of the joint system is given by: χ in W (z) = exp( 1 zt s α s z s ) exp( 1 zt Eα E z E ) (74) We work in the Heisenberg picture, hence the change brought about by the channel action will be on the position and momentum operators of the joint system i.e. ˆR and not ˆρin. The channel action is summarized as follows: Let ˆR denote the column comprising of the transformated position and momentum operators of joint system i.e. the ones after the channel action. Then ˆR is related with ˆR by (53). In detail, this looks like the following: 18 From this point upto the point before classification of various channels, z represents the phase-space variables of the joint system i.e. the principal system and the environment whereas z s represents the phase space variables of the principle system only 3

25 ˆq s ˆp s ˆq E ˆp E ˆq s = X Y ˆp s X Y ˆq E ˆp E X ˆq s + Y ˆq E ˆp s ˆp E = X ˆq s + Y ˆq E ˆp s ˆp E (75) Hence we have obtained the transformed position and momentum operators for the joint system. The new position and momentum operators will now be used in the displacement operators i.e. the displacement operators are changed to: ( ) ˆD (z) = exp[ι ( ˆq s,ˆp s ) X T + ( ˆq E,ˆp E ) Y T, ( ˆq s,ˆp s ) X T + ( ˆq E,ˆp E ) Y T z s ] = exp[ι (ˆq s,ˆp s ) (X T z s + X T z E ) + ι (ˆq E,ˆp E ) (Y T z s + Y T z E )] (76) z E The characteristic function for the joint system changes as follows: χ in W (z) = χ s W (X T z s + X T z E )χ E W (Y T z s + Y T z E )) (77) After the channel action suppose we want to pick out the characteristic function of the principal system only, we set the phase space variables of the environment system to zero. Upon so doing we obtain: χ s W (z s ) = χ s W (X T z s )χ E W (Y T z s ) = exp[ 1 zt s (Xα s X T )z s ] exp[ 1 zt s (Y α E Y T )z s ] (78) Our channel is basically identified by two things: the symplectic matrix which enacts the transformation on the joint system, S and the environment sytem (determined by the covariance matrix α E ). Specifying both of these uniquely identifies the Gaussian channels. To bring the Gaussian channel in a canonical form, 4

26 what we do is to apply symplectic transformations on the principal system before and after the channel has been applied to it. The enactment of these is to be considered as part of the whole channel. We see that by these symplectic transformations, every one-mode Gaussian channel (for a one-mode environment) can be brought to one canonical form or the other. This is done in the following way: Perform an Sp(, R) transformation, T 1 on the ˆq s and ˆp s operators before the channel action. T 1 ˆq s T 1 α s T1 T (79) ˆp s (79) tells us basically that instead of applying S to the joint system comprising of the principal system (as it was given to us) and the environment, we change the principle system by applying a local symplectic transformation on it and apply S on this joint system. Next, perform an Sp(, R) transformation, T on the ˆq s and ˆp s operators after the channel-action. T ˆq s = T (X ˆq s + Y ˆp s ˆp s ˆq E ) T T z s (80) ˆp E Hence we can see that performing T 1 on the principal system after the channel action is alternately equivalent to transforming the principal system phase space variables by T1 T. (80) basically tells us that we change the outpute state. Using both of these new features in the pre-existing channel (78) becomes: χ s W (z s ) = exp[ 1 zt s T (XT 1 α s T T 1 X T )T T z s ] exp[ 1 zt s (T Y α E Y T T T )z s ] = χ s W (T 1 X T T T z s )χ E W (Y T T T z s ) (81) Having specified the procedure to obtain the canonical forms for a Gaussian state, we now proceed to the classification. Case A1:X = 0 and hence according to (7),detY = 1 So Y Sp(, R) i.e. Y ΩY T = Ω. Since detx = 0 the completely-positive condition: α ι Ω 0 is trivially satisfied since α is a covariance matrix for the environment system. The change in the characteristic 5

27 function for the system thus is 19 : χ s W (z) χ s W (z) = χ E W (z) = exp( 1 zt T Y α E Y T T T z) (8) Now since Y Sp(, R), α = Y α E Y T is also a covariance matrix for the environment. Since we want the simplest form of the channel, we choose T such that in accordance with Williamson s theorem T αt T is symplectic diagonal. In that case T αt T = (N )I. Hence we get: χ s W (z) = exp( 1 (N )(x + y )) (83) Since X = 0, Y = 0 and X Sp(, R). Hence our symplectic matrix in canonical form looks like: S = 0 I Q 0 (84) where Q Sp(, R) and where the environment system is a thermal Gaussian state with the mean number N 0. We still have a degree of freedom over what P can be but for the simplest form we choose it to be I. A1 is called the completely depolarizing channel. All information of the input is lost after the application of this channel. Case A: X is a rank 1 matrix. dety = 1. The completely-positive conditions are the same as before and is again trivially satisfied. The only difference is in the canonical form adopted by the channel. One can apply local symplectic transformations before and after the application of the channel to bring the symplectic matrix in the following canonical form using (68),(69) and (70): S = c 11 c 1 0 c 1 (85) c 1 c 0 c where the environment system is again a thermal system with the mean number N 0. We still have degree of freedom to choose what X and Y should be. X = c 11 c 1 Sp(, R). c 1 c 19 For convenience, z = z s represents the phase space variabls of the principal system again from here onwards 6

28 Our characteristic functions changes as follows: χ s W (z) = χ s W (x, 0)exp( 1 (N )(x + y )) (86) Both A1 and A are referred to as Singular channels because X matrix is rank 0 and 1 respectively. Case B1: detx = 1 and Y = 0. The positivity condition is just α 0 which we know holds. Applying suitable local symplectic transformations before and after the application of the channel the symplectic transformation comes to the form: S = I 0 0 Q (87) where Q Sp(, r). This is the ideal channel which doesn t change the state of the system after running it through the channel. It adds no noise to the input state. The system is not effected by any environment in this case. Case B: detx = 1 and Y is rank 1. The positivity condition is again α 0 which holds. Using local symplectic transformations before and after the application of the channel gives us: S = c 11 0 c 11 c 1 (88) c 1 0 c 1 c Of course, Y Sp(, R). The characteristic function change is: χ s W (z) = χ s W (z)exp( 1 N cy ) (89) We have the degree of freedom of choosing N c 0. Since the noise is embedded in only one quadrature this is known as the Single Quadrature Channel. Case C: detx = k 0. Then dety = 1 k. The completely-positivity conditions is α ι k 1 Ω 0. Hence we get that α should be a bonafide covariance matrix for the environmental system. Again by local k 1 symplectic transformations we can choose a canonical form for S: 7

29 C1: When 0 k 1 (68),(69) and (70) impose the condition: detx = 1 k and dety = k. Any X and Y which satisfy these are a suitable candidate for the respective lower sub-matrices of S and we have the freedom to choose X and Y upto the said conditions being satisfied. S = k 0 1 k 0 0 k 0 1 k 1 k 0 k k 0 k (90) where the environment system is again a thermal state with mean number N 0. Characteristic function change: χ s W (z) = χ s W (kz)exp( 1 (1 k )(N )(x + y )) (91) C1 is the beam-splitter channel with additional Gaussian noise due to interaction with a thermal bath. This can be seen from the following effect on the canonical variables of the principal system: q s p s = q 1 k kq E p 1 k kp E (9) We see that the effect of the channel amounts to an SO() between the position operators of the principal system and the environment and the SAME rotation between the corresponding momentum operators of the principal system and environment. This channel is also known as the attenuator channel since the mean number N of the output is lower than the input. This is shown as follows: Let the covariance matrix of the system be α s before the application of C1(k). Then N = T r(α s ) 1. After C1(k) has been applied, N = k (T r(α s ) 1) 0. Since 0 k 1, we see that the mean number reduces as a consequence of this channel. C: When k 1 (68),(69) and (70) impose the condition: detx = 1 k and dety = k. Any X and Y which satisfy these are a suitable candidate for the respective lower sub-matrices of S and we have the freedom to choose 0 This is for the case where N 0 = 0. Hence the thermal environment is in a vacuum state. 8

30 X and Y upto the said conditions being satisfied. S = k 0 k k 0 k 1 k 1 0 k 0 0 k 1 0 k (93) where the environment system is again a thermal state with mean number N 0. Characteristic function changes as: χ s W (z) = χ s W (kz)exp( 1 (k 1)(N )(x + y )) (94) This is the amplification channel. As in the mean number N increases since it goes from T r(α) 1 to k (T r(α) 1). Since k 1 this time, the mean number increases. Case D: detx = k and hence dety = k + 1 The completely positive condition is that matrix. α + ι Ω 0. Hence α 1+k k +1 has to be a bonafide covariance (68),(69) and (70) impose the condition: detx = 1 + k and dety = k. Any X and Y which satisfy these are a suitable candidate for the respective lower sub-matrices of S and we have the freedom to choose X and Y upto the said conditions being satisfied. Using local symplectic transformations the canonical form of the symplectic transformation effecting the channel S = k 0 k k 0 k + 1 k k 0 0 k k (95) where the environment system is again a thermal state with mean number N 0. Characteristic function change: χ s W (ξ) = χ s W (k x y )exp( 1 (k + 1)(N )(x + y )) (96) This channel is known as the phase-conjugation channel or the transposition channel. The reason for 9

31 this is that undergoing this channel means that the operator gets transposed with additional Gaussian noise required to maintain the channel completely positive. Here we have obtained the canonical forms of Gaussian Channels. While A and B channels don t have any channel parameter to identify them, C and D channels require two parameters to identify them completely. One is k which gives the degree of attenuation/amplification and the other N 0 which specifies the temperature of the environment which one is working with. The quantum limited channels (channels with just the minimum amount of noise to keep the channel completely positive) are exempted from this particular parameter. 5.1 Kraus Representation: One-Mode Bosonic Gaussian Channels One can obtain the Kraus representation for one-mode bosonic gaussian channels 1. The Kraus representation for all the channels are obtained for a 0 temperature thermal environment.but later using the semi-group property we can see that the concatenation of different quantum limited channels give us the Kraus representation for noisy channels (at higher temperatures) too. Let ˆρ s and ˆρ E = 0 0 be the respective initial states of the system and environment. Let ˆρ s be the state of the system after the channel has acted upon it. Let Û be the unitary evolution of the joint system during the channel action.as in (5) trace the environment out. This is done in the Fock state basis here: ˆρ s = T r E (Û ˆρ s 0 E 0 Û ) (97) = l E Û 0 E ˆρ s 0 E Û l E = l=0 l=0 Ŵ l ˆρ s Ŵ l where Ŵl represent the Kraus Operators. Consider expanding Ŵl in the Fock state basis : Now Ŵ l = n 1,m 1 C m1l n 1 0 m 1 n 1 (98) C m 1m n 1 n = m 1 m Û n 1n = dx 1 dx m 1 m x 1 x x 1 x Û n 1 n (99) We see that the matrices in (84),(85),(87),(88),(90),(93) and (95) GL(, R) GL(, R) subgroup 1 J Solomon Ivan, R. Simon, K. Sabapathy Since the fock state basis will be utilized here, it makes more sense to work with the complex representation featuring â and â operators than the ˆp and ˆq operators 30

32 of Sp(4, R) where the position operators of the principal and environment system intermingle amongst themselves and same so for the momentum operators. All the symplectic transformations can, thus, be brought to the form: S = M (M 1 ) (100) where M acts on the position operators of both modes only and (M 1 ) acts on the momentum operators of both modes only. Since Û corresponds to a symplectic transformation on the canonical variables, x 1x Û can be easily evaluated: x 1, x Û = (M 1 (x 1, x ) T ) 1, (M 1 (x 1, x ) T ) = x 1, x (101) Hence in (99) we get: C m 1m n 1 n = dx 1 dx ψm1(x 1 )ψm(x 1 )ψ n1 (x 1)ψ n (x ) (10) where ψ n (x) is the wavefunction of the n-fock state. Utilizing the generating function for the wave function of Fock states in the equation above we get: C m 1,m 1 n 1,n = 1 n1!n!m 1!m! m 1 η m 1 1 m η m n 1 z n 1 1 n z n where F (z 1, z, η 1, η ) is the generating function and is given by F (z 1, z, η 1, η ) z1,z,η 1,η =0 (103) F (z 1, z, η 1, η ) = 1 π dx 1 dx exp{ 1 [(x 1 η 1 ) + (x η ) (104) + (x 1 z 1 ) + (x z ) η 1 η z 1 z ]} (105) 5. Singular Case:A 5..1 A1 In the case of A1 we don t need to use the technique given above, the Kraus operators can be deduced directly as: 31

33 Ŵ l = 0 l (106) Then: A1( ˆρ s ) = 0 l ˆρ s l 0 = 0 0 (107) i=0 Some properties of the channel are that it s non-classicality breaking 3 and hence entanglement breaking 4. Also it has a fixed point at ˆρ s = 0 0. All operators of the form Ŵ l Ŵ l are mutually orthogonal. This proves that this operator is a an extremal channel. 5.. A The symplectic matrix is given by: S = (108) Here M = 1 0 and M 1 = We can use the generating function given by equation (103) but in place of the fock state in the place of m, we expand using the position eigen-kets q : where q represents the position eigenvalue. The Kraus operators come to be: where q ) represents the coherent state α with α = q. C m 1,q n 1,n = m 1, q Û n 1, m 1 (109) ˆV q = C m1,q n 1,0 m 1 n 1 = q ) q (110) m 1,n 1 3 A channel being non-classicality breaking implies that for whatever input to the channel, the output will necessarily be classical 4 A channel being entanglement breaking implies that for whatever input state the output of channel will necessarily be separable. 3

34 These Kraus operators can be easily verified by checking consistency with (86) 5. The trace preserving property of these Kraus operators is easily seen to be satisfied. Given that this channel has a representation of rank one Kraus operators it is non-classicality breaking and hence entanglement breaking 6. This channel has no fixed points. Now, the set of all operators { ˆV q ˆV q } is a mutually orthogonal set which means that the given channel is an extremal channel. 5.3 B B1 This is the identity channel i.e. the channel doesn t change the state of the system at all. This is obviously not non-classicality breaking, not entanglement breaking either and every point is a fixed point. The Kraus operators are given by identity matrices. This channel is obviously not an extremal channel B This is the identity channel along with some Gaussian noise in one quadrature. Hence this is not a quantum-limited channel. Just as in the case of A channel we can find the Kraus operators by using the position eigenkets instead of the fock-state eigenkets. The Kraus operators are given by: Ẑ q = 1 (N c π) 1 4 exp[ q N c ] ˆD( q ) (111) This channel is neither entanglement breaking nor non-classicality breaking nor is it extremal 5.4 C C1 Attenuator Channel The generating function (104) in this case becomes: F (z 1, z, η 1, η ) = exp[η ( 1 k z 1 + kz ) + η 1 (kz 1 1 k z ] (11) Utilizing the generator function,we get the Kraus operators to be: ˆB l (k) = m+l C l ( 1 k ) l k m m m + l, l = 0, 1,... (113) m=0 5 N 0 = 0 in this case 6 Entanglement Breaking Channels in Infinite Dimensions-A.S. Holevo (008) 33