Entanglement or Separability an introduction

Transcription

1 Bachelor Thesis Entanglement or Separability an introduction Lukas Schneiderbauer December 22, 2012 Quantum entanglement is a huge and active research field these days. Not only the philosophical aspects of these spooky features in quantum mechanics are quite interesting, but also the possibilities to make use of it in our everyday life is thrilling. In the last few years many possible applications, mostly within the Quantum Information field, have been developed. Of course to make use of this feature one demands tools to control entanglement in a certain sense. How can one define entanglement? How can one identify an entangled quantum system? Can entanglement be measured? These are questions one desires an answer for and indeed many answers have been found. However today entanglement is not yet fully in control by mathematics; many problems are still not solved. This paper aims to provide a theoretical introduction to get a feeling for the mathematical problems concerning entanglement and presents approaches to handle entanglement identification or entanglement measures for simple cases. The reader should be aware of the fact that this paper constitutes in no way the claim to be a summary of all available methods, there exist many more than demonstrated in the following pages. Student ID number Degree course Physics assisted by ao. Univ.-Prof. i.r. Dr. Reinhold Bertlmann

2 Contents 1 Introduction 3 2 Preliminary definitions Composite quantum systems Density operator Reduced density operator for a bipartite quantum system Entangled states A pure correlated composite state Hilbert-Schmidt space Qudit systems Qubit systems Pure bipartite qudit states Schmidt decomposition Von-Neumann entropy Mixed bipartite qudit states A criterion for non-entanglement Generalization of the Von-Neumann entropy Requirements for entanglement measures Entanglement measures Generalization of the Schmidt rank Entanglement witnesses Entanglement Witness Theorem (EWT) Positive Map Theorem (PMT) Positive Partial Transpose (PPT) Criterion Bertlmann-Narnhofer-Thirring Theorem Multipartite states Some entanglement measures Geometric measure of entanglement Measure of entanglement by Barnum An entanglement witness The choice of factorization The choice of factorization for pure states The choice of factorization for mixed states Conclusion 24 References 24

3 1 INTRODUCTION 1 Introduction Quantum entanglement, named by Erwin Schrödinger 1, is a quantum mechanics phenomenon which was first outlined by Albert Einstein, Boris Podolski and Nathan Rosen in 1935 and led to the famous EPR paradoxon.[6] Their result was that quantum mechanics can t be a correct theory since entanglement would lead to a violation of the classical principle of local realism:»we are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete.«- Einstein, Podolski, Rosen [6] On the other hand quantum mechanics was succussfully confirmed by experiments. Therefore the idea of hidden variables arose, meaning quantum theory is not fundamental but a statistical theory which covers the fundamental theory. This makes it possible to deny a violation of local realism as matter of principle and at the same time to believe in the success of the theory John Stewart Bell provided a way to determine experimentally whether hidden variables exist or not (see his Bell inequalities[4]). Surprisingly the experiments turned out to deny the existence of such hidden variables. Although the interpretation of quantum theory may vary, today quantum mechanics is widely accepted as fundamental theory by physicists. This work is an introduction developing criteria to identify entangled quantum systems for specific cases. To start at an uniform level it first provides some fundamental mathematical definitions in section 2. Mathematical entities with physical meaning like the Hilbert space or a density operator are defined. Also a (mathematical) answer to the question»what is entanglement?«is given. The following derivations are based on these definitions. Next it faces one of the simplest non trivial problems in section 3: a pure bipartite qudit state. Two important concepts, the Schmidt-decomposition and the Von-Neumann entropy, are introduced which will prove to be useful in further studies. The Bell states will function as examples. In chapter 4 mixed bipartite qudit systems are discussed. First a simple but important criterion for non-entanglement is derived. Generalization approaches of the Von-Neumann entropy and the Schmidt rank are presented as entanglement measures which will include the entanglement of formation, the relative entropy of entanglement, the entanglement of distillation, entanglement cost and a Hilbert- Schmidt measure. Also the concept of entanglement witnesses with a few applications like the Positive Partial Transpose Criterion is displayed. Furthermore a small outlook for multipartite systems is given in section 5. Two more measures of entanglement are presented and a specific entanglement witness for an N-partite system is given. And last but not least the issue of the possibility to choose different algebra factorizations with respect to the consequences to entanglement is discussed in chapter 6. 1 The original German name for this spooky feature was»verschränkung«. 3

4 2 PRELIMINARY DEFINITIONS 2 Preliminary definitions 2.1 Composite quantum systems A quantum system is represented by a Hilbert space H. denoted by H A, H B and so forth. Let s consider a number of such systems, Definition 2.1. It is postulated that the composite system of these subsystems H A, H B,... represented by their Product-Hilbert space H AB... is H AB... = H A H B... (2.1) An operator O in a composite system S AB... is denoted by O AB.... For a product state Ψ A Ψ B... one also writes Ψ A, Ψ B, Density operator Definition 2.2. Given an ensemble { ϕ i, p i } of N possible pure states ϕ i with probability p i one defines the density operator ρ N N ρ := p i ϕ i ϕ i, p i = 1 (2.2) Note that the ϕ i are not the eigenstates of ρ, therefore not orthogonal in general. This density operator ρ represents a mixed state 2 (see e.g. [2]) of a quantum system and has the following important properties: ϕ ρ ϕ 0 ϕ H ρ = ρ (2.3) tr(ρ) = 1 (2.4) Proof. Eq. (2.3) is obvious and eq. (2.4) can be shown by just using the definition of the trace: tr(ρ) i Ψ i ρ Ψ i with an arbitrary ON basis {Ψ i } Reduced density operator for a bipartite quantum system Definition 2.3. Let ρ AB be a density operator on H AB. Then the reduced density operator ρ A is defined as ρ A := tr B (ρ AB ), (2.5) whereas tr B (Z AB ) is called a partial trace, which is defined by tr B (Z AB ) := n ΨB n Z AB Ψ B n with arbitrary basis { Ψ i }. The result is an operator on H A. More to partial traces can be found in [2]. The reduced density operator ρ A can be envisioned as the state in the subsystem S A. All probability predictions for local measurements (the related observable is in the form of A I) on system S A can be allocated to the reduced density operator. 2 A mixed state is a generalization of a pure state. To see this, set N = 1 in eq. (2.2) and one gets the density operator of a pure state, namely ρ pure = ϕ ϕ. 4

5 2.3 Entangled states 2 PRELIMINARY DEFINITIONS 2.3 Entangled states Definition 2.4. A composite state is called correlated if and only if its density operator ρ AB... can not be written as a product operator, i.e. ρ AB... ρ A ρ B... (2.6) Since ρ represents a mixed state in general, correlation alone may not necessarily imply a deviation from classical views. To classify a non classical effect we go on with further definitions. Definition 2.5. A state is called separable if and only if the density operator ρ AB... can be written as ρ AB... = n p r ρ A r ρ B r... (2.7) r=1 Note, that for n = 1, ρ AB... is not correlated. The state for n 1, that is a separable correlated state, is called a classical correlated state. From the definition it is clear that the family of separable states is a convex set 3. Definition 2.6. A state is called entangled if and only if the state is correlated and not separable, i.e. ρ AB... n p r ρ A r ρ B r... (2.8) It may be useful to take a look at a special case: a pure and correlated state A pure correlated composite state r=1 Claim. A pure correlated composite state is an entangled one. Proof. Let ρ ρ AB... be the density operator of a pure state Ψ Ψ AB..., i.e. ρ = Ψ Ψ. Choose a vector Φ, so that Ψ Φ = 0. First, we make the attempt to decompose ρ to a convex sum 4 of other density operators: ρ = = Φ ρ Φ = 0 = n λ r ρ r r=1 n λ r Φ ρ r Φ r=1 Since all Φ ρ r Φ are positive (see eq. (2.3)) and λ r are positive, the above equation holds, if and only if Φ ρ r Φ = 0. Now complete Ψ to an orthonormal basis { φ k }, φ 1 Ψ and look at the matrix elements of ρ and ρ r in this basis: φ i ρ r φ j. We conclude, that all elements vanish, except for Ψ ρ Ψ = Ψ ρ r Ψ = 1, because of eq. (2.4). Therefore ρ = ρ r r [1; n]. Thus a decomposition is not possible. Furthermore let s assume, ρ is correlated, i.e. ρ ρ 1 ρ 2..., then ρ cannot be written in the form of eq. (2.7). Hence it must be entangled. That implies: Ψ AB... Φ A Φ B... Ψ AB... entangled (2.9) 3 A set C is said to be convex if (1 t) x + t y C, (x, y) C, t [0,1]. 4 The convex sum of ρ is ρ = n r=1 λrρr, n r=1 λr = 1, λr > 0 5

6 2.4 Hilbert-Schmidt space 2 PRELIMINARY DEFINITIONS 2.4 Hilbert-Schmidt space Definition 2.7. Let H be a Hilbert space describing a quantum mechanical system and H the set of operators acting on H. Then the set H forms a Hilbert space for itself. This space is called a Hilbert-Schmidt space. The scalar product on A denoted by B, C (B, C H) is defined by and induces the Hilbert Schmidt norm B := B, B. 2.5 Qudit systems B, C := tr(b C) (2.10) Definition 2.8. A quantum system with d linear independent states is called Qudit system and it is described in a d-dimensional Hilbert space H d Qubit systems A qubit system is an important specific case of a qudit system, namely a system with d = 2. The system is represented by a 2-dimensional Hilbert space H 2. To name a few physical examples: Spin particle Polarization of single photons Quantum dots Bell states It is not the purpose to discuss Bell states in detail here, however, it makes a good example for an entangled bipartite qubit state as we will see later on. Definition 2.9. Let 0 and 1 be the eigenstates of the Pauli operator 5 σ 3 acting on H 2, then we define the so called Bell basis also known as Bell states: Φ ±, ψ ± H AB = H A 2 HB 2. Φ ± := 1 2 ( 0, 0 ± 1, 1 ), ψ ± := 1 2 ( 0, 1 ± 1, 0 ) (2.11) One could calculate its reduced density operators (see eq. (2.5)) which are in the form: ρ A = ρ B = 1 2 I (2.12) 5 Since the Pauli operators are not the subjects here, a detailed description is omitted, but can be referred in e.g. [2]. 6

7 3 PURE BIPARTITE QUDIT STATES 3 Pure bipartite qudit states After the basic but important definitions we will start discussing pure bipartite systems. A very important concept is the so called Schmidt decomposition which can be applied to all pure bipartite states, sadly only for those. In preparation for more complicated matters, e.g. mixed states, the Von-Neumann entropy and its properties will be discussed. 3.1 Schmidt decomposition For a pure bipartite qudit state Ψ AB H d it is useful to introduce a decomposition named the Schmidt decomposition, bi-orthogonal expansion or polar expansion. (see [7]) Theorem 3.1. Let Ψ AB be a normed state in the composite system S AB in the Product-Hilbert-Space H AB = H A H B with dimension dim H A = a and dim H B = b. The reduced density operators of the subsystems S A and S B are given by ρ A = tr B (ρ AB ) and ρ B = tr A (ρ AB ) with the density operator ρ AB = Ψ AB Ψ AB. Then Ψ AB can be written in the form (3.1) with normed eigenstates u A n, wn B and eigenvalues p A n = p B n of ρ A and ρ B, i.e. ρ A u A n = p n u A n, p A n R ρ B w B n = p n w B n, p B n R Ψ AB = Proof. First, expand Ψ AB into the basis of eigenstates of ρ A and ρ B : k pn u A n, wn B, p n > 0 (3.1) n=1 a,b Ψ AB = i,j u A i, wj B Ψ AB u }{{} =:c ij A i, w B j (3.2) The reduced density operator ρ A can be written as a decomposition of his eigenstates: { a ρ A = p n u A n u A > 0 0 n k a n, p n = 0 k + 1 n a, p n = 1 (3.3) n=1 On the other side ρ A must fulfill its role as a reduced density operator: ρ A = tr B (ρ AB ) = tr B ( Ψ AB Ψ AB ) 3.2 = = = a,b i,j a,b k,l n=1 a,b a,b {}}{ c ij c kl tr B ( u A i, wj B u A k, wb l ) i,j k,l b c ij c kl wm w B j B wl B wb m u A i u A k m a,b a,b c ij c kl wl B wb A j ui u A k }{{} = a,b i,j k,l i,j δ ij u A i ua k wb j wb l a c ij c kj u A i u A k (3.4) By comparison of eq. (3.3) and (3.4) one finds that c mn = 0 for m n and c 2 nn = p A n. After the very similar calculation for ρ B, one gets c 2 nn = p B n. Thus p A n = p B n p n. With these results it can immediately be seen that eq. (3.2) reduces to eq. (3.1). k 7

8 3.2 Von-Neumann entropy 3 PURE BIPARTITE QUDIT STATES Definition 3.1. The number k in eq. (3.1) is called the Schmidt rank of Ψ AB, the eigenvalues {p n } are called Schmidt coefficients. Claim. Ψ AB is not entangled if and only if the Schmidt rank k = 1. Proof. We know from eq. (2.9), that a pure state is entangled, if and only if the state is correlated. Since { u A n } and { w B n form a basis, Ψ AB can only be written in the form (2.9) if k = 1. Claim. k = 1 tr((ρ A ) 2 ) = tr((ρ B ) 2 ) = 1. Proof. k = 1 means, all eigenvalues {p n } of ρ A = ρ B ρ vanish except of one. The trace of ρ is 1 (see eq. (2.4)). Now suppose k > 1. We know, p n > 0. Then p n : p n < 1 = p 2 n < p n. Therefore the trace tr((ρ ) 2 ) = k n p2 n < 1. Hence the trace tr((ρ ) 2 ) = 1 k = 1. Let us summarize this important and useful statement: For a pure bipartite qudit state one can recognize entanglement by evaluating the trace of the squared reduced density operators. What we have found is a nice criterion to detect entanglement. Another interesting side product of the Schmidt decomposition is the gained knowledge, that a subsystem of a pure entangled state cannot be pure (remember that a pure state means rk(ρ A ) = 1 and this is equal to k = 1). Example. Schmidt decomposition and Bell states Using the above statement and the results from eq. (2.12) we can directly calculate the trace of the reduced Bell states: tr((ρ A ) 2 ) = tr( 1 4 I) = 1 2 This finding is indeed interesting. It tells us, that the Bell states must be entangled. 3.2 Von-Neumann entropy Another important tool to get hints about existent entanglement is the Von-Neumann entropy.[1] Definition 3.2. The Von-Neumann entropy S(ρ) is defined as S(ρ) := tr(ρ log ρ) (3.5) One can think of it as the quantum version of the entropy Ŝ known in thermodynamics. To make this clear, one can for example show that the Von-Neumann entropy of the density matrices of the micro canonical, the canonical and the grand canonical ensemble coincides with the entropy Ŝ. As the thermodynamic entropy, the Von-Neumann entropy is a measure of information. For practical reasons it is often useful to express S in the eigenvalues of ρ: Claim. The Von-Neumann entropy S(ρ) can be written in the form d S(ρ) = (λ i log λ i ) d λ i = 1 (3.6) with the eigenvalues λ i of ρ. 8

9 3.2 Von-Neumann entropy 3 PURE BIPARTITE QUDIT STATES Proof. Evaluate the trace with respect to eigenbasis {ψ i } of ρ: which is the claimed formula. S(ρ) = tr(ρ log ρ) = tr(log ρ ρ) = d d ψ i log ρ ρ ψ i = λ i ψ i log ρ ψ i = d λ i log λ i ψ i ψ i = }{{} =1 d λ i log λ i The Von-Neumann entropy S(ρ) has among others the following properties: 1. The entropy of a pure state is its minimum value. 2. The entropy of a density operator with rank d fulfills ρ = Ψ Ψ S(ρ) = 0 (3.7) 0 S(ρ) log d (3.8) Proof. To see the first property, one recognizes that ρ has only one eigenvalue λ 1 = 1 if and only if ρ is a pure state. With this finding one can evaluate eq. (3.6), which tells S(ρ pure ) = 0. Property 2 can be proven by calculating extremal values of function (3.6). We need to include the constraint i λ i = 1 in the discussion, therefore we rewrite the expression as d d 1 λ i = 1 = λ i + λ d d 1 = λ d (λ i d ) = 1 λ i and thus the entropy reads With λ d λ l d 1 S(λ l d ) = λ i log λ i p d (λ l d ) log p d (λ l d ) = 1 the derivate of S(λ l d ) emerges as S d 1 = ( λ i log λ i + λ i ) λ d log λ d λ d λ l λ l λ l λ l λ l = log λ l 1 + log λ d + 1 = log λ l + log λ d This is zero if λ max l = λ d = λ max l = 1 d, hence the local maximum Smax is determined. S max = S( 1 d ) = d 1 d log 1 d = log d Since S max is the only local maximum (on the whole domain including the boundaries) it is a global one. The global minimum value S min = 0 is obvious (all 0 λ i 1). 9

10 3.2 Von-Neumann entropy 3 PURE BIPARTITE QUDIT STATES We already know from section 2.3.1, that a pure state can only possess non classical correlations (or none at all). Let s take a look at the valuable example of a pure qubit state. Example. Von-Neumann entropy and Bell states Let s calculate the entropy for our example, the Bell states ρ AB and its reduced operator ρ A = ρ B. To find S(ρ AB ), we can use property (3.6) of the entropy S; since ρ AB is pure per definition: S(ρ AB ) = 0 This result shows no new findings, it just confirms the obvious: The pure Bell state is a state with maximum information (as every pure state is of course). The reduced density operator ρ A apparently possesses the eigenvalues { 1 2, 1 2 }. We use eq. (3.6) and get the entropy: S(ρ A ) = log 2 = log d With property (3.8) one notices that this is the maximum value of the entropy. As a consequence, the sub state ρ A is a maximal undetermined state. With these considerations on an information theory level, one is tempted to define a measure of entanglement: Definition 3.3. Consider a pure state Ψ AB with the corresponding density matrix ρ AB and reduced density matrices ρ A and ρ B. Then define the entropy of entanglement E(ρ AB ) = E(Ψ) 0 E(Ψ) := S(ρ A ) = S(ρ B ) 1 (3.9) and call it a measure of entanglement of the state Ψ AB. A state with E(Ψ) = log d, d = dim H is called maximal entangled (see the Bell state). This is the first step to the more general approach of entanglement measures as we will see in chapter 4. Such measures also play a very important role for mixed states. 10

11 4 MIXED BIPARTITE QUDIT STATES 4 Mixed bipartite qudit states As mixed states describe a more complicate matter than pure states, there are less concrete criteria available. This chapter begins with a derivation of a specific criterion for non-entanglement. Then entanglement measures are introduced as a generalization of the Von-Neumann entropy. After a renewed discussion about the Schmidt rank we will proceed with establishing a more geometrical approach by the so called entanglement witnesses. 4.1 A criterion for non-entanglement In section 3.1 we have seen that a sub-state 6 cannot be pure if its pure super-state is entangled. We will generalize this now.[2] Theorem 4.1. Let S AB be a bipartite qudit system described in H AB = Hd A HB e and let ρ AB be a mixed density operator acting on H AB in this system. The corresponding reduced density operator of the subsystem S A is ρ A. Furthermore if ρ A is pure then the state ρ AB is separable ( not entangled). Proof. The mixed density operator ρ AB can be written in its spectral basis: ρ AB = s n Ψ AB n Ψ AB n, s n = 1, s n > 0 (4.1) n n For simplicity set Ψ AB n Ψ n. Now expand Ψ n to the spectral basis of ρ A and ρ B : {u i } and {w j }: (cp. eq. (3.2)) Ψ n = c n,ij u i, w j (4.2) i,j Using the expansion from eq. (4.2), eq. (4.1) reads ρ AB = n s n c n,ij c n,kl u i, w j u k, w l }{{} i,j k,l u i u k w j w l (4.3) To find some constraints for the coefficients c n,ij, we evaluate the reduced density operator ρ A in an arbitrary ON basis { ϕ r }: ρ A = tr B (ρ AB ) = n = n { }}{ s n c n,ij c n,kl ϕ r w j w l ϕ r u i u k i,j k,l r = w l w j =δ ij s n c n,ij c n,kj u i u k (4.4) i,j k Because { u i } is the eigenbasis of ρ A, it can also be written as: ρ A = n r n u n u n (4.5) 6 It may be a slight abuse of the used vocabulary, nevertheless it s a good aberration of state of a subsystem. 11

12 4.2 Generalization of the Von-Neumann entropy 4 MIXED BIPARTITE QUDIT STATES By comparison of eq. (4.4) and eq. (4.5), c n,ij = 0 for i j. Suppose now, that ρ A is a pure state, i.e. ρ A = u 1 u 1. Hence all c n,ij are zero except c n,11. This implies that eq. (4.3) reduces to ρ AB = n s n c 2 n,11 ( u 1 u 1 w 1 w 1 ) (4.6) which is the form of a separable state (see eq. (2.7)). To sum up, this means: If a system is in a pure state it cannot be entangled with another system. 4.2 Generalization of the Von-Neumann entropy We have seen in section 3.2 that for pure bipartite states the Von-Neumann entropy is a measure of entanglement. For mixed states this measure fails to distinguish classical and quantum correlations. There are however some attempts to generalize the Von-Neumann entropy. The so called entropy measures are designed to coincide on the special case of pure bipartite states and to be equal to the entropy of the reduced density operators. What succeeds now are the requirements for a general entanglement measure[5] followed by definitions of some specific cases.[5],[16] Note that not all of these provide a rule to do a practical evaluation. Generally the calculation might not even be possible Requirements for entanglement measures The following properties for a good entanglement measure E are of course not mandatory and the discussion for such requirements is still open. Actually some of the measures introduced below do not fulfill all of the following properties. 1. ρ is separable E(ρ) = 0 2. Normalization for an entangled state of a d-dimensional system. 0 E(ρ) log d (4.7) 3. LOCC (local operations and classical communication) cannot increase E Continuity E(ρ) E(σ) 0 for ρ σ 0 (4.8) 5. Additivity: n identical copies of the state ρ should contain n times the entanglement of one copy. 8 E(ρ n ) = n E(ρ) (4.9) 6. Subadditivity: E(ρ σ) E(ρ) + E(σ) (4.10) 7. Convexity: The entanglement measure should be a convex function, i.e. with 0 < λ < 1. E(λρ + (1 λ) σ) λ E(ρ) + (1 λ) E(σ) (4.11) 7 A measure which decreases under LOCC is called entanglement monotone. 8 A m := A A... A }{{} m times 12

13 4.2 Generalization of the Von-Neumann entropy 4 MIXED BIPARTITE QUDIT STATES Entanglement measures Entanglement of formation Definition 4.1. Let ρ AB = i p iρ i, ρ i ϕ i ϕ i be a mixed bipartite state, then the i-th entropy of entanglement E i E(ρ i ) for the pure states ρ i is given by eq. (3.9), i.e. E(ρ i ) = S(tr B (ρ i )). The entanglement of formation also known as entanglement of creation is then defined as E F (ρ AB ) := min {dec} where min {dec} means the minimum over all possible decompositions of ρab. p i E i (4.12) To express this formula in words: Form the average entanglement entropy for pure states over a decomposition. Do this with all possible decompositions and pick the minimum. Relative entropy of entanglement Definition 4.2. Let ρ AB be a mixed bipartite state (see eq. (2.7)) and σ AB a separable one in the system S AB. Then one defines the relative entropy of entanglement E R (ρ AB ). E R (ρ AB ) := i min (log ρ AB log σ AB )] (4.13) σ AB S ABtr[ρAB To understand the structure of this formula let s compare it with the original entropy S(ρ) = tr(ρ log ρ). E R looks like tr(ρ log ρ σ ) 9. So one could speak of this as the entropy of a state relative to an arbitrary separable one. As above, form this entropy for all available separable states and take the minimum. Entanglement of distillation Proposition. Suppose ρ A is the reduced density operator of ρ AB. For such a ρ A one can find a pure super state φ AB H A H B, where ρ A is again the corresponding reduced density operator. This process is called purification. It makes clear that a reduced density operator is not unique. The statement follows from the Schmidt decomposition theorem in section 3.1. Definition 4.3. The entanglement of distillation E D (ρ AB ) tells us about the number m out of maximal entangled states φ AB that can be purified from n in copies of a given state ρ AB (written as (ρ AB ) n in ). More precisely it is defined as the ratio m out over n in in the limit of infinitely many states (ρ AB ) n in. E D (ρ AB ) := m out lim (4.14) n in n in A mixed state, that can be distilled, i.e. E D > 0 is called free entangled. Its counterpart is bound entanglement, for E D = 0. Entanglement cost In contrast to the entanglement of distillation the entanglement cost quantifies the amount of pure-state entanglement needed to create ρ AB using local operations. Definition 4.4. The entanglement cost E C (ρ AB ) is therefore defined as E C (ρ AB ) := m in lim (4.15) n out n out where m in is the number of maximal entangle states φ AB required to prepare (ρ AB ) nout. 9 Mathematically this expression is of course only valid in the sense of definition (4.13). 13

14 4.3 Generalization of the Schmidt rank 4 MIXED BIPARTITE QUDIT STATES The entanglement cost has not been computed for any mixed state yet, however the calculation can be done for our examples, the Bell states.[17] Example. Let ρ p a mixture of the two Bell states Φ + and Φ (see eq. (2.11)), i.e. Then the entanglement cost E C is given by ρ p (1 p) Φ + Φ + + p Φ Φ p [0; 1 2 ] E C (ρ p ) = H 2 ( p (1 p)) where H 2 (x) = S (x, 1 x) with Shannon entropy S (p 1, p 2 ) = i p i log p i. 10 Remarkable is that E D and E C are extremal measures, any sensible entanglement measure should lie in between. Hilbert-Schmidt measure Similar to the relative entropy of entanglement we can pursue an even more geometrical approach by using the Hilbert-Schmidt distance ρ 1 ρ 2. Definition 4.5. For an arbitrary state ρ AB the Hilbert-Schmidt measure is defined as where S is the set of all separable states in H A H B. E H (ρ AB ) := min ρ AB σ AB (4.16) σ AB S It is intuitively clear that E H serves as a measure of entanglement since it evaluates the distances to all separable states and takes their minimum. As will be shown in chapter the Hilbert-Schmidt measure plays an important role connecting entanglement measures with entanglement witnesses. 4.3 Generalization of the Schmidt rank Similarly to the generalization approach for the entanglement of formation (4.2.2), one can define the Schmidt number as generalization of the Schmidt rank (discussed in section 3.1).[5] Definition 4.6. Let ρ AB = i p iρ i, ρ i ϕ k i i ϕk i i be a mixed bipartite state. Every pure state ρ i has its Schmidt rank k i. Set k max max k i. The Schmidt number r is defined as the minimum of the highest Schmidt rank over all possible decompositions. r := min {dec} k max (4.17) The Schmidt number r cannot be higher than the smaller of the dimensions of the two subsystems. It is called a measure of entanglement and useful in connection with Schmidt witnesses which are discussed below. 4.4 Entanglement witnesses A quite geometrical approach is the use of so called entanglement witnesses. Entanglement witnesses cover a huge area by themselves. Again this chapter is not meant to be a summary of the whole topic but specific topics are chosen. A central role plays of course the Entanglement Witness Theorem (EWT).[11] 10 Proofing this would need a couple of preparations and is omitted. 14

15 4.4 Entanglement witnesses 4 MIXED BIPARTITE QUDIT STATES Entanglement Witness Theorem (EWT) The Entanglement Witness Theorem states the following: Theorem 4.2. A state ρ ent H is entangled if and only if there exists a Hermitian operator A H which satisfies ρ ent, A < 0 σ, A 0 σ S (4.18) where S is the set of all separable states. A is then called an entanglement witness. This can be derived via the Hahn-Banach Theorem[12] of functional analysis. A geometrical representation of this theorem states the following:[10] Theorem 4.3. Let C be a convex closed set in a topological vector space X, and let b X but b / C. Then there exists a hyperplane that separates b from the set C. Proof. of Theorem 4.2. If we identify the above vector spaces X H and sets C S we know according Theorem 4.3 that there exists a hyperplane which separates entangled and separable states. Since such a hyperplane must exist there exists an A H which satisfies ρ, A = 0 where ρ is an arbitrary vector in H. Theorem 4.2 follows straightaway with figure 4.1a (exchange A by A if necessary). Definition 4.7. An entanglement witness A is called optimal, that is A opt, if there exists a separable state σ S such that σ, A opt = 0 (4.19) It is called optimal since it detects more entangled states than normal witnesses would do as figure 4.1b illustrates. (a) A plane in Euclidean space (b) An optimal entanglement witness. Figure 4.1: Geometrical illustration of entanglement witnesses, Source: [10] Positive Map Theorem (PMT) From the EWT another useful theorem can be derived, namely the Positive Map Theorem (PMT): Theorem 4.4. Let Λ be a positive map acting on the Hilbert space. Then a bipartite state ρ is separable if and only if (I Λ)ρ 0 positive maps Λ. (4.20) 15

16 4.4 Entanglement witnesses 4 MIXED BIPARTITE QUDIT STATES Proof. The =- direction is shown easily.[10] One applies (I Λ) to a separable state 2.7 and gets (I Λ)ρ = i p i ρ A i Λρ B i. Since Λ is positive, Λρ B i is positive as well and therefore (I Λ)ρ is positive. The proof for the = - direction is more complex and provided by Michael, Pawel and Ryszard Horodecki [8]. In other words that means if we can find a positive map Λ where (I Λ)ρ < 0, i.e Λ is not completely positive 11, we know that ρ is entangled. Example. An often stated example is the partial transposition PT : (I T ). Let s test it on the Bell state Φ + (see eq. (2.11)). The matrix representation of its corresponding density operator ρ B Φ + Φ + and its partial transposition reads (I T )ρ B = (I T ) = Calculating the eigenvalues leads to { 1 2, 1 2, 1 2, 1 2 } which means (I T )ρ B is not positive, therefore ρ B is entangled (as we already know) Positive Partial Transpose (PPT) Criterion Theorem 4.4 can be used to create a more specific criterion for the special case of a bipartite qubit system. It is a necessary and sufficient criterion developed by Peres and Horodecki [8]. Theorem 4.5. A state ρ acting on H 2 H 2, H 2 H 3 or H 3 H 2 is separable if and only if its partial transposition ρ T B is a positive operator, i.e. Definition 4.8. A state satisfying eq. (4.21) is called a PPT state. ρ T B (I T )ρ 0. (4.21) The fact that this is indeed a necessary condition can immediately be seen recalling the PMT (Theorem 4.4) and the stated example of the partial transposition. As shown in [10], to prove that the criterion is also sufficient, one needs a theorem by Stormer and Woronowitz [14],[19] which is repeated here for the sake of completeness. Theorem 4.6. Any positive map Λ that maps operators on Hilbert spaces H 2 H 2, H 2 H 3 or H 3 H 2 can be decomposed in the following way where Λ CP 1 and Λ CP 2 are completely positive maps. Λ = Λ CP 1 + Λ CP 2 T (4.22) Proof. Let ρ be a state which satisfies (I T )ρ 0. Since Λ CP 1 and Λ CP 2 are completely positive it must be true that (I Λ CP 1 )ρ + (I Λ CP 2 )(I T )ρ 0 11 A positive map Λ is called completely positive if the map (I d Λ) is still a positive map d. 16

17 4.4 Entanglement witnesses 4 MIXED BIPARTITE QUDIT STATES which can obviously be written as (I (Λ CP 1 + Λ CP 2 T ))ρ 0 (4.23). Now the need of Theorem 4.6 gets apparent. It tells us that eq. (4.23) reduces to (I Λ)ρ 0 where Λ is a positive map. Now since any Λ can be decomposed in terms of (4.22) the PMT (theorem 4.4) can be used which tells us that ρ is separable if and only if (I T )ρ Bertlmann-Narnhofer-Thirring Theorem The Bertlmann-Narnhofer-Thirring Theorem connects the Hilbert-Schmidt measure defined in section with the concept of entanglement witnesses. More specific it tells that the Hilbert-Schmidt measure of an entangled state equals the maximal violation of an inequality which basically defines the entanglement witnesses. Before going into details we will proof a Lemma which will considerably ease the proof of the theorem itself. After that the mentioned inequality will be defined and the theorem will be stated. Lemma. Let ρ 1 and ρ 2 be two arbitrary states and an operator C be defined as 12 C(ρ 1, ρ 2 ) := ρ 1 ρ 2 ρ 1, ρ 1 ρ 2 I ρ 1 ρ 2 (4.24) Also let ρ 0 be the nearest separable state of ρ ent, i.e. ρ 0 ρ ent = min σ S σ ρ ent where S denotes again the set of all separable states, then C := C(ρ 0, ρ ent ) is an optimal entanglement witness of ρ ent as defined by eq. (4.18) and (4.19). Proof. In order to be an entanglement witness of ρ ent the operator C has to fulfill ρ ent, C < 0 and σ, C 0 σ S. That this is satisfied can be shown in the following manner: Looking at figure 4.2 it is obvious that ρ p ρ 1, defines a hyperplane through ρ 1 orthogonal to ρ 1 ρ 2. By evaluating ρ, C(ρ 1, ρ 2 ), which is ρ, C(ρ 1, ρ 2 ) = ρ, ρ 1 ρ 2 ρ 1 ρ 2 ρ 1, ρ 1 ρ 2 ρ 1 ρ 2 = 0 ρ 1 ρ 2 ρ, I = ρ ρ ρ 1 ρ 2 }{{} 1, =1 one recognizes that the above hyperplane definition reduces to 13 ρ 1 ρ 2 ρ 1 ρ 2 ρ p, C = 0 (4.25), similar one defines the left-hand and right-hand states ρ l and ρ r (see figure 4.2) as: ρ l, C < 0 ρ r, C > 0 12.,. is defined by eq. (2.10). 13 In this context omitting the arguments of C(.,.) means evaluating C(ρ 1, ρ 2). 17

18 4.4 Entanglement witnesses 4 MIXED BIPARTITE QUDIT STATES Figure 4.2: A hyperplane orthogonal to ρ 1 ρ 2, Source: [10] Now consider C C(ρ 0, ρ ent ). Since ρ 0 is the nearest separable state of ρ ent it lies on the boundary of S. Therefore the plane defined by ρ p, C = 0 is tangent to S. Also the plane is orthogonal to ρ ent ρ 0. So we have ρ ent, C < 0, σ, C 0 σ S and ρ p, C = 0. Thus C is indeed an optimal entanglement witness. It is not hard to see that the definition (4.18) of an entanglement witness can be rewritten as σ, A ρ ent, A 0 σ S (4.26) This inequality is also called a generalized Bell inequality (GBI). One can define the maximal violation of the GBI as follows. Definition 4.9. The maximal violation of the GBI is given by B(ρ ent ) := max A, A ai 1 ( min σ S ( σ, A ρ ent, A ) ) (4.27) where a is the coefficient corresponding to the identity matrix when expanding A into the Pauli-Matrix basis {σ i σ j }. However, the specific choice of the used norm will not be of any importance to us. Theorem 4.7. The Hilbert-Schmidt measure E H (ρ ent ) of an entangled state ρ ent equals the maximal violation of the GBI B(ρ ent ): E H (ρ ent ) = B(ρ ent ) (4.28) Proof. The distance ρ 1 ρ 2 of two states can be written as ρ 1 ρ 2 = ρ 1 ρ 2, ρ 1 ρ 2 ρ 1 ρ 2 = ρ ρ 1 ρ 2 1 ρ 2, ρ 1 ρ 2 + c I where c is a complex constant because (ρ 1 ρ 2, c I) = c (tr(ρ 1 ) tr(ρ 2 )) = 0 since tr(ρ) = 1 ρ. Choosing c = ρ 1,ρ 2 ρ 2 ρ 1 ρ 2 yields per definition to ρ 1 ρ 2 = ρ 1 ρ 2, ρ 1 ρ 2 ρ 1, ρ 1 ρ 2 I ρ 1 ρ 2 = ρ 1, C(ρ 1, ρ 2 ) ρ 2, C(ρ 1, ρ 2 ) (4.29) Using eq. (4.29) the Hilbert-Schmidt measure E H (ρ ent ) can be written as: E H (ρ ent ) = min σ S ρ ent σ = ρ ent ρ 0 = ρ 0, C ρ ent, C = 18

19 4.4 Entanglement witnesses 4 MIXED BIPARTITE QUDIT STATES From the definition of an optimal entanglement witness it is clear that max A ρ ent, A = ρ ent, A opt (A opt has to be parallel to ρ ent ) and ρ 0, A opt = 0 since ρ 0 is on the plane. Since C is an optimal entanglement witness we can rewrite E H (ρ ent ) further as ( ) = max ( ρ 0, A) ρ ent, A ) = max min ( σ, A) ρ ent, A ) = B(ρ ent ) A A σ S which completes the proof. 19

20 5 MULTIPARTITE STATES 5 Multipartite states The classification and quantification of entanglement is in its early stage when it comes to multipartite systems. Many measures are designed to function in specific domains where they perform successful operations. At the same time the same measures might fail badly in other areas. The mathematical treatment of such measures is a delicate matter. The next few sections provide the definitions of more or less often stated entanglement measures. [1] 5.1 Some entanglement measures Geometric measure of entanglement An attempt to quantify the entanglement of a pure state is given by the minimal distance of the state from the set of all pure product states Φ. Definition 5.1. A geometric measure E g (Ψ) for a pure multipartite state is defined as follows: E g (Ψ):= log max Φ Ψ Φ 2 (5.1) As shown in [18] this measure can be extended to an entanglement monotone measure for mixed states. It is zero for separable states and rises up to its maximum for e.g. the maximal entangled N-particle GHZ state Measure of entanglement by Barnum Another measure was provided by Barnum.[3] Definition 5.2. A generalized entanglement measure for qubit systems is given by E gl := 2 2 N N tr(ρ 2 j) (5.2) j=1 where N is the number of qubits. ρ j is the reduced density matrix of the j-th qubit. Remembering the consequence of the Schmidt decomposition for pure bipartite states (section 3.1), which stated that tr((ρ A ) 2 ) = 1 for non entangled states, eq. 5.2 looks like an average of such evaluations. However, a deeper connection would remain to be shown. 5.2 An entanglement witness The concept of entanglement witnesses can in principle be extended to general multipartite systems since the proof done in section is independent of the dimension of the system. Of course, in practice it is quite hard to find useful quantities. In [9] the hermitian operator W in an N-partite system is studied: W = 1 2 k {0,1} N b k (Q + k Q k ) (5.3) where Q ± k are projectors to generalized GHZ states, i.e. Q ± k = G ± k G ± k with G ± = 1 ( k ± σ k x N ) k 2 (5.4) and b k are constrained coefficients. 20

21 5.2 An entanglement witness 5 MULTIPARTITE STATES It turns out that under the constraint (5.5) the hermitian operator W witnesses entanglement. b k 2 N (5.5) k By choosing different sets of b k one can obtain specific W s for different purposes. 21

22 6 THE CHOICE OF FACTORIZATION 6 The choice of factorization Until now when we spoke e.g. about a bipartite qubit system in a 4-dimensional Hilbert space H 2 H 2 we silently ignored the specific structure of these tensor factors. Mostly it was implied that the local measurement is a projective measurement to the 0 state, which has of course its meaning, because that is how it s mainly done in experiments. However, that is certainly a special case. In general it is allowed to do measurements w.r.t. all states in the eigenspace of a particular Hamilton operator that relates to the actual present physical problem. We will lift this restriction now and study what would happen to states (respectively entanglement) if we chose different factorization of the tensor algebra. Among other things we will find that for an entangled state we can always find a factorization of its algebra in which this same state does not appear to be entangled but not the other way around! 6.1 The choice of factorization for pure states For pure states the situation is quite clear as can be seen in the following section.[15] Theorem 6.1. For any pure state ρ H with H being a d 1 d 2 -dimensional Hilbert-Schmidt space one can find a factorization of H, H = A1 A 1, such that ρ is separable w.r.t. this factorization as well as a factorization H = B 1 B 2 such that ρ is maximal entangled. Proof. To see this it suffices to know that for any Ψ H there exists a unitary operator U which transforms Ψ in the following way: U Ψ = 1 Ψ 1,i Ψ 2,i (6.1) d i with d = min(d 1, d 2 ). Let s consider Ψ to be separable w.r.t. H = A1 A 1. So with unitary transformations we can transform every separable state in a maximal entangled one. Since ρ sep = Ψ Ψ the transformed state has the form ρ ent = Uρ sep U A 1 A 1. Instead of the state ρ we can as well transform the underlying algebra: B B 2 = U(A 1 A 1 )U. Now it becomes clear that ρ which appears to be separable w.r.t H = A 1 A 1 is maximal entangled w.r.t H = B 1 B The choice of factorization for mixed states For mixed states the situation is not that easy and in fact much more interesting. To proof theorem 6.2 we need a maximal entangled ONB for a d d-dimensional Hilbert space. The state (6.2) provides one which is proven in lemma 6.1.[15] Lemma 6.1. The set of vectors χ kl = j e 2πi d jl ϕ j ψ j+k, j = 1,..., d; k = 1,..., d (6.2) is a maximal entangled ONB of the d d-dimensional Hilbert space H d H d. Proof. Evaluating χ kl χ mn yields to χ kl χ mn = i,j = i,j e 2πi d jl e 2πi d in ( ϕ j ψ j+k ) ( ϕ i ψ i+m ) e 2πi = δ km d (lj ni) ϕ j ϕ i }{{}}{{} j ψ j+k ψ i+m =δ ij =δ (j+k),(i+m) e 2πi d j(l n) = δ km δ ln 22

23 6.2 The choice of factorization for mixed states 6 THE CHOICE OF FACTORIZATION which proves that the set of χ kl s is an ONB. That these states are entangled can be seen by considering the Schmidt decomposition (see section 3.1). The Schmidt rank equals its maximum value. Theorem 6.2. For any mixed state ρ H with H being a d 1 d 2 -dimensional Hilbert-Schmidt space one can find a factorization of H, H = A1 A 1, such that ρ is separable w.r.t. this factorization. A factorization H = B 1 B 2 such that ρ is entangled exists only beyond a certain bound of mixedness. Proof. To realize the first statement one has to recall that ρ can be written as α p α χ α χ α where χ α can be an arbitrary state. We ve already mentioned that every pure state can be transformed into a separable state with a unitary transformation U: U χ α = ϕ i,α ψ j,α. Transforming ρ yields to UρU = α p α ϕ i,α ϕ i,α ψ j,α ψ j,α (6.3) which is separable. On the other hand we can choose a unitary transformation U which transforms every state in a maximal entangled one: U χ α = χ kl where χ kl = j e 2πi d jl ϕ j ψ j+k (see Lemma 6.1). The transformed state ρ is a so called Weyl state: UρU = k,j p k ϕ j ϕ j ψ j+k ψ j+k (6.4) This is the expansion of UρU into a set of maximal entangled states. But this set is not convex, therefore the state has not necessarily to be entangled. Knowing this it makes sense to introduce the following definition: Definition 6.1. If a state ρ remains separable for any factorization of its algebra (that is for any unitary transformation U of ρ) it is called a absolutely separable state. In [15] this theorem is stated more precisely for e.g. Werner states ρ w in d d dimensions: ρ w = αp + 1 α d 2 I d 2 0 α 1 (6.5) where P is a projector to a maximal entangled state. This state is entangled for α > 1 d+1. It is shown that if the largest eigenvalue λ 1 of ρ w satisfies λ 1 > 3 then there exists always an algebra such that d 2 ρ w appears entangled. Furthermore the following theorem has been proven for general mixed states in 2 2 dimensions: Theorem 6.3. Let ρ be an arbitrary mixed state in 2 2 dimensions with an ordered spectrum {λ i }, i = 1,.., 4 with λ 1 λ 2 λ 3 λ 4. If the eigenvalues satisfy then ρ is absolutely separable. λ 1 λ 3 2 λ 2 λ 4 0 (6.6) 23

24 7 CONCLUSION 7 Conclusion Let s summarize the previous steps. We started studying the simplest non trivial case, a pure bipartite state. For these states concrete methods are available to exactly identify entanglement. Looking at the Schmidt decomposition Ψ AB = k n=1 pn u A n, wn B we derived that one just has to evaluate the trace of the squared reduced density operator to detect entanglement. Also the Schmidt rank quantifies entanglement in a certain sense. It was shown that the Von-Neumann entropy S(ρ) = tr(ρ log ρ) as generalization of the entropy Ŝ from thermodynamics (and therefore leading to a measure of information) offers useful properties by considering sub systems and was our first candidate for an entanglement measure. In mixed bipartite states we generalized the idea of the entropy and defined a general entanglement measure to not only detect entanglement but to quantify it. We have seen that there is no unique measure and stated a few possibilities, each for different purposes. Further we saw that pure systems cannot be entangled with other systems. In this sense the whole is not the sum of its parts but more:»bestmögliches Wissen um ein Ganzes schließt nicht notwendig das Gleiche für seine Teile ein.«- Erwin Schrödinger[13]»The best possible knowledge of a whole does not necessarily include the best possible knowledge of all its parts.«from the Hahn-Banach Theorem of functional analysis it follows that there must exist an operator which detects entanglement, namely an entanglement witness, which was discussed as well. An operational criterion, the PPT-criterion, was derived with the help of these witnesses. It was shown that one is even able to construct an optimal entanglement witness that was used to proof the Bertlmann-Narnhofer-Thirring Theorem which connects the concept of entanglement witnesses with the approach of entanglement measures. The quantification of multipartite states is even more of a challenge. Mathematically the most measures are not fully understood in the sense that many properties are just proven for special cases which are merely assumed to hold in general. Not only the quantification is a problem: the question, if a state is actually entangled or not, cannot be answered in general. Initial attempts were made to provide tools, but that hopefully should only be the beginning. Finally the factorization of the algebra was considered. The results were quite interesting. We can transform every entangled state to a non-entangled one while the reverse is not true in general but just for certain bound of mixedness. 24

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