Maximal Entanglement A New Measure of Entanglement

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1 1 Maximal Entanglement A New Measure of Entanglement Salman Beigi School of Mathematics, Institute for Research in Fundamental Sciences IPM, Tehran, Iran arxiv: v1 [quant-ph] 11 May 014 Abstract Maximal correlation is a measure of correlation for bipartite distributions. This measure has two intriguing features: 1 it is monotone under local stochastic maps; it gives the same number when computed on i.i.d. copies of a pair of random variables. This measure of correlation has recently been generalized for bipartite quantum states, for which the same properties have been proved. In this paper, based on maximal correlation, we define a new measure of entanglement which we call maximal entanglement. We show that this measure of entanglement is faithful is zero on separable states and positive on entangled states, is monotone under local quantum operations, and gives the same number when computed on tensor powers of a bipartite state. I. MAXIMAL CORRELATION Suppose that two parties, Alice and Bob, respectively receive a A and b B sampled from a joint distribution p. Their goal is to respectively output c C and d D with some predetermined distributionq CD. The question is whether this goal is achievable, assuming that there is no communication link between Alice and Bob. We call this problem the local transformation problem transformation of a bipartite distribution to another one under local stochastic maps. Deciding whether local transformation is possible, is a hard problem in general due to the non-linearity of the problem, especially when the size of A B is large. In this case one is willing to obtain necessary or sufficient conditions for the problem. Local stochastic maps channels cannot transform a less-correlated bipartite distribution, to a morecorrelated one. So we may find bounds on this problem using measures of correlation. For instance, due to the data processing inequality, if the mutual information between A and B, denoted by IA;B, is less than IC;D then p cannot be transformed to q CD under local operations. Suppose that Alice and Bob receive multiple samples from the source distribution p, and want to generate only one sample from q CD. That is, their inputs are a n A n and b n B n, for some arbitrary n, sampled with probability p n a n,b n := i pa i,b i, and they want to output c and d with probability qc,d. Mutual information in this case does not give any bound simply because IA n ;B n = nia;b is greater than IC; D for sufficiently large n assuming that IA; B is non-zero. In general, additive or even weakly additive measures of correlation do not give any bound for the latter problem. There is a measure of correlation called maximal correlation which is first introduced by Hirschfeld [1] and Gebelein [] and then studied by Rényi [3], [4]. Later it is studied in [5] by Witsenhausen, and in the recent works [6], [7], [8], [9], [10], [11]. Maximal correlation is defined as follows: µp = max E[fg] 1 E[f] = E[g] = 0, E[f ] = E[g ] = 1, in which f and g are functions on A and B respectively. µp is indeed the maximum of the expectation value of the product of two random variables that have zero mean and variance 1. Maximal correlation is not additive. Indeed this measure of correlation, has the intriguing property that µp n A n B n = µp. Moreover, as a measure of correlation it satisfies the data processing inequality. Putting these together we conclude that if µp < µq CD then local transformation of p to p CD is impossible even when an arbitrary large number of samples of the source p is available. Let us consider an example. Let A = B = and define the distribution p ǫ p ǫ ab = { 1 ǫ a = b, ǫ a b. by Then for 0 ǫ 1/ we have µ p ǫ = 1 ǫ. As a result, having even infinitely many samples from p ǫ we cannot locally generate samples from pδ if 0 δ < ǫ 1/.

2 Maximal correlation has recently been defined for bipartite quantum states [1]. In this paper we first review the definition and properties of maximal correlation in the quantum case. Then based on maximal correlation, we define a new measure of entanglement, which we call maximal entanglement. We show that this measure of entanglement is zero on separable states and positive on entangled ones, is monotone under local quantum operations, and gives the same number when computed on tensor powers of a bipartite state. This measure of entanglement, however, is not monotone under classical communication. II. MAXIMAL CORRELATION FOR QUANTUM STATES Let us first fix some notations. The Hilbert space corresponding to a register A is denoted by H A, and the space of linear operators acting on H A by LH A. In this paper we use Dirac s ket-bra notation for vectors: vectors of H A are denoted by v, and vectors in the dual space are denoted by w ; Then v w is a linear operator and belongs to LH A. The dimension of H A is denoted by d A in this paper we only consider finite dimensional quantum registers. The same notations are adopted for other quantum registers. We equip LH A with the Hilbert-Schmidt inner product X,Y := trx Y for X,Y LH A, where X is the adjoint of X. Now to define maximal correlation for a bipartite density matrix ρ LH A LH B, we should think of f, g in 1 as quantum observables, whose expectations are computed with Born s rule. Based on this intuition, maximal correlation of ρ is defined by µρ = max trρ X A Y B trρ A X A = trρ B Y B = 0, 3 trρ A X A X A = trρ BY B Y B = 1. 4 Here ρ A = tr B ρ and ρ B = tr A ρ are the reduced density matrices on subsystems A and B respectively, and X A LH A and Y B LH B. As shown in [1] in the above optimization one may assume that X A and Y B are hermitian. To study properties of µρ it would be useful to define ρ = I A ρ 1/ B ρ ρ 1/ A I B. Here the inverses of ρ A and ρ B are defined on their supports Moore-Penrose pseudo-inverse. In fact without loss of generality, by restricting H A,H B to the supports of ρ A,ρ B respectively, we may assume that the local density matrices are full-rank. The following theorem proved in [1] is the quantum version of the results of [7] and [9] connecting maximal correlation to certain Schmidt coefficients see also [11]. Theorem 1 [1] The first Schmidt coefficient of ρ as a vector in the bipartite Hilbert space LH A LH B is equal to 1, and its second Schmidt coefficient is equal to µρ. Note that LH A and LH B are Hilbert spaces equipped with the Hilbert-Schmidt inner product. So Schmidt decomposition, and then Schmidt coefficients, of vectors in LH A LH B are computed with respect to Hilbert-Schmidt inner product. To be more precise, the Schmidt decomposition of ρ is of the form ρ = λ i M i N i, i where λ 1 λ 0 are Schmidt coefficients, and M i LH A and N i LH B are orthonormal bases, i.e., trm i M j = trn i N j = δ ij where δ ij is the Kronecker delta function. The above theorem states that λ 1 = 1 and λ = µρ. Let us examine this theorem in the classical case. Let p be a joint distribution. Let P be a A B -matrix whose a,b-th entry is equal to p ab. Also let P A be the diagonal matrix with diagonal entries p a. Define P B similarly. Define P = P 1/ A P P 1/ B which is the classical analogue of ρ defined above. Then the first singular value of P is equal 1 and its second singular value is equal to µp. For example, if A = B = {0,1}, then P = p 00 p00+p 01p 00+p 10 p 10 p10+p 11p 00+p 10 p 01 p00+p 01p 01+p 11 p 11 p10+p 11p 01+p 11.

3 3 P has two singular values one of which is 1. So the other one is equal to µp = det P. 5 Relating maximal correlation of ρ to Schmidt coefficients of ρ, we can now state the main properties of maximal correlation. Theorem [1] µ satisfies the following properties: a µρ σ A B = max{µρ,µσ A B }. b Let Φ A A,Ψ B B be completely positive trace-preserving super-operators. Let σ A B = Φ Ψρ. Then µσ A B µρ. Proof: a This is a simple consequence of the fact that Schmidt coefficients of the tensor product of two vectors are equal to the pairwise products of Schmidt coefficients of the two vectors. b Completely positive trace-preserving maps are compositions of isometries and partial traces. It is not hard to see that local isometries do not change µρ. So we only need to show that µσ µσ AA BB. This inequality holds because in the definition of maximal correlation for σ AA BB we may restrict X AA,Y BB to have the form X AA = X A I A and Y BB = Y B I B. With these restrictions we obtain µσ as the optimal value. The following corollary is a direct consequence of the above theorem. Corollary 1 Suppose that ρ n, for some n, can be locally transformed to σ EF. Then µρ µσ EF. Let us workout an example. Let ψ = be the Bell state on two qubits. Define ρ ǫ = 1 ǫ ψ ψ +ǫ I 4, 6 where 0 ǫ 1 and I /4 is the maximally mixed state. It is not hard to see that µ ρ ǫ = 1 ǫ see [1] for details. Now using the above corollary, having an arbitrary large number of copies of ρ ǫ one cannot transform them into one copy of ρ δ under local transformations if ǫ > δ. The following theorem characterizes the extreme values of maximal correlation. Theorem 3 [1] 0 µρ 1 and the followings hold: a µρ = 0 if and only if ρ = ρ A ρ B. b µρ = 1 if and only if there exist local measurements {M 0,M 1 } on A, and {N 0,N 1 } on B such that 0 < trρ M 0 N 0 < 1, and trρ M 0 N 1 = trρ M 1 N 0 = 0. c For a pure state ρ = ψ ψ, we have µρ = 0 if ψ is a product state, and µρ = 1 if ψ is entangled. Part b of this theorem is the quantum version of Witsenhausen s result [5] that maximal correlation of a bipartite distribution is equal to 1 iff the distribution has a common data. We finish this section by proving two lemmas which will be used in the next section. Lemma 1 Letρ be a bipartite density matrix such that ψ ρ ψ 1 ǫ, where ψ = Then µρ 1 ǫ. Proof: There is nothing to prove for ǫ 1/ since maximal correlation is non-negative. So we assume that 0 ǫ < 1/. Let us define ψ i = i 11, i = 0,1 ψ i = i 10 i =,3. Then ψ 0 = ψ and { ψ i : 0 i 3} is an orthonormal basis for the space of two qubits. Therefore there are c ij C such that 3 ρ = c ij ψ i ψ j. i,j=0

4 4 By assumption we have c 00 1 ǫ. Moreover, since ρ is a density matrix i c ii = 1 and c 01 c 00 c 11, c 1 c c Now suppose that we measure each qubit A,B in the computational basis { 0, 1 }. We let p EF be the outcome distribution. We have p 00 = 1 c 00 +c 11 +c 01 +c 10 p 11 = 1 c 00 +c 11 c 01 c 10 p 01 = 1 c +c 33 +c 3 +c 3 p 10 = 1 c +c 33 c 3 c 3. By the monotonicity of maximal correlation under local measurements Theorem we have µρ µp EF. Moreover, maximal correlation of p EF can be computed using 5. Thus we obtain µρ p 00 p 11 p 01 p 10 [. p00 +p 01 p 00 +p 10 p 10 +p 11 p 01 +p 11 ]1 Let q = c 00 +c 11 which gives c +c 33 = 1 q, a = c 01 +c 10 and b = c 3 +c 3. Then q 1 ǫ, and by 7 we have a q and b 1 q. Indeed there is a stronger upper bound on a ; Using c 00 1 ǫ and c 00 +c 11 = q we have a c 01 c 00 c 11 1 ǫq 1+ǫ. Using this inequality, and the fact that q 1 ǫ 1/ we find that a q 1. Now rewriting the above bound in terms of these new variables we obtain µρ = = q a 1 q +b [ 1+a+b1+a b1 a+b1 a b ] 1/ q 1+b a [ 1+a4 +b 4 a b a b ] 1/ q 1+b a [ 1+a4 +b 4 a b a b ] 1/, where in the last line we use a q 1. The numerator of the latter bound is obviously an increasing function of b. Moreover using the fact that b 1 q 1 + a, it is not hard to see that the denominator is a decreasing function of b. Therefore q 1 a µρ 1 a q 1 = 1+ 1 a 1+q 1 1 ǫ. Lemma Let {σ n } be a sequence of density matrices and suppose that lim n σ n = ρ and that lim n µσ n = λ. Then we have µρ λ. In particular if λ = 0, then ρ is a product state. Proof: Let X,Y be the operators that achieve the optimal value µρ in -4. For every n define and X n := Y n := X trσ n A [, n tr σ A XX trσ n A X ] 1/ Y trσ n B [. n trσ B YY trσ n B Y ] 1/

5 5 Then for every n we have trσ n A X n = trσ n B Y n = 0 and trσ n A X nx n = trσ n B YY = 1. On the other hand, it is easy to see that lim tr σ n n X n Y n = tr ρ X Y = µρ. This means that if µρ λ+ǫ for some ǫ > 0, then for sufficiently large n, µ σ n λ+ǫ/, which is a contradiction since lim n µσ n = λ. Therefore µρ λ. Note that, by the assumption of the above lemma we cannot conclude the equality of µρ and λ. For example, define the distributions p n by pn 00 = 1 1/n, pn 11 = 1/n and pn 01 = pn 10 = 0. Then for every n we have µp n = 1, but this sequence of distributions converges to q with q 00 = 1 and q 01 = q 10 = q 11 = 0, for which we have µq = 0. III. MAXIMAL ENTANGLEMENT In this section we define a measure of entanglement in terms of maximal correlation. By part c of Theorem 3, maximal correlation is already a measure of entanglement on pure states; This is the measure that is 0 for pure product states and 1 for pure entangled states. Nevertheless, maximal correlation is not a measure of entanglement since there are bipartite classical distributions whose maximal correlation is 1. Convex roof extension is the idea that is usually applied to construct measures of entanglement in such situations. In our case however, convex roof does not result in a measure that satisfies our desired properties. So we propose the following definition: µ ent ρ := infmaxµτ i i, where the infimum is taken over all decompositions ρ = i p i τ i where p i 0 and τ i s are density matrices. We call µ ent maximal entanglement. From the definition we clearly have i 0 µ ent ρ 1. ii µ ent ρ µρ. iii µ ent ρ = 0 for all separable states ρ. iv If ρ is pure then µ ent ρ = µρ. v Maximal entanglement is quasi-convex, i.e., we have µ ent i λ iρ i maxi µ ent ρ i. Statement iii holds simply because every separable state can be written in the form ρ = i p iτ i τ i B, and for all i we have µτi A τi B a unique decomposition. The following theorems state the main properties of maximal entanglement. A = 0. Statement iv holds because pure states essentially have Theorem 4 Suppose that σ A B = Φ Ψρ where Φ A A and Ψ B B are completely positive trace-preserving maps. Then µ ent ρ µ ent σ A B. Proof: Let ρ = i p iτ i be a decomposition of ρ. Then σ A B = i p iφ Ψτ i is a decomposition of σ A B. Using Theorem for every i we have µ τ i i µ Φ Ψτ. Then taking maximum over i, and then infimum over all decompositions of ρ we obtain the result. Theorem 5 For all density matricesρ,σ A B we haveµ entρ σ A B = max{µ entρ,µ ent σ A B }. Proof: Using the data processing inequality Theorem 4 by considering local partial trace maps tr and tr A B we obtain µ ent ρ σ A B max{µ entρ,µ ent σ A B }. For the other direction consider decompositions ρ = i p iτ i and σ A B = j q jξ j A B. Then ρ σ A B = i,j p i q j τ i ξj A B,

6 6 is a decomposition of ρ σ A B. As a result µ ent ρ σ A B maxµ τ i i,j = max max i,j ξj A B { µ τ i,µ ξ j A B }. Taking infimum over all decompositions of ρ and σ A B gives the desired result. The above two theorems imply the following analogue of Corollary 1 for maximal entanglement. Corollary Suppose that ρ n, for some n, can be locally transformed to σ EF. Then µ ent ρ µ ent σ EF. Theorem 6 Maximal entanglement is faithful, i.e., µ ent ρ = 0 if and only if ρ is separable. Proof: We already know that separability of ρ implies µ ent ρ = 0. So we need to show that if µ ent ρ = 0 then ρ is separable. By definition µ ent ρ = 0 means that for every ǫ > 0 there is a decomposition ρ = i p iτ i such that µτ i ǫ for all i. Let S ǫ be the set of all density matrices whose maximal correlation is at most ǫ: S ǫ := {σ : µσ ǫ}. Therefore, µ ent ρ = 0 implies that ρ is in the convex hull of S ǫ for every ǫ > 0. Now using Carathéodory s theorem, for every ǫ > 0 there is a decomposition ρ = M i=1 p ǫ i τ i,ǫ, such that µ τ i,ǫ ǫ for all i, and M = MdA,d B is a constant independent of ǫ. Now by a standard compactness argument there is a sequence ǫ 1 > ǫ > with lim n ǫ n = 0 such that p ǫn 1,...,p ǫn M,τ1,ǫn,...,τM,ǫn, converges to some p 0 1,...,p0 M,τ1,0,...,τM,0. This tuple then gives a valid decomposition where {p 0 i ρ = M i=1 p 0 i τ i,0, is a density matrix. On the other hand, since lim n τ i,ǫn = τi,0 and that µτi,ǫn ǫ n, using Lemma we find that µτ i,0 = 0. Then by part a of Theorem 3, τ i,0 is a product state for every i. As a result ρ is separable. : 1 i M} is a probability distribution and each τ i,0 We showed in Theorem 4 that maximal entanglement is monotone under local quantum operations. As a measure of entanglement one may expect that maximal entanglement is also monotone under classical communication too. However, this is not the case. Theorem 7 Maximal entanglement is not monotone under classical communication. Proof: We know that any two-qubit entangled state is distillable [13]. That is, having copies of an entangled two-qubit stateρ, approximations of the maximally entangled state ψ = can be distilled using local quantum operations and classical communication LOCC. However, we know that there are two-qubit entangled states ρ such that µ ent ρ = µ ent ρ n µρ < 1 = µ ent ψ ψ see for instance the example after Corollary 1. On the other hand, local quantum operations do not increase maximal entanglement. Therefore, it should be classical communication that sometimes increases maximal entanglement. Computing maximal entanglement µ ent ρ seems a hard problem in general since we need to take an infimum over all decompositions of ρ. Finding upper bounds on µ ent ρ however, is easy; We only need to pick a decomposition to find an upper bound. In particular we have µ ent ρ µρ.

7 7 In the following we present some ideas that may serve as a useful tool for proving lower bounds on maximal entanglement. In the previous section we observed that the maximal correlation of ρ ǫ defined by 6 is equal to µ ρ ǫ = 1 ǫ. Computing the maximal entanglement of these states however, does not seem easy. By the partial transpose test we know that ρ ǫ is separable if and only if ǫ /3 [14], [15]. Then using the faithfulness of maximal entanglement Theorem 6, µ ent ρ ǫ = 0 if ǫ /3 and µ entρ ǫ > 0 otherwise. ǫ In the following we present a characterization of µ ent ρ in terms another optimization problem. Theorem 8 Define λǫ = min{µσ : ψ σ ψ 1 3ǫ/4}, where ψ = 1 ǫ ǫ Then µ ent ρ = λǫ, where ρ = 1 ǫ ψ ψ + ǫi /4. Moreover, λǫ = 0 for ǫ /3, and for ǫ < /3 we have 1 3ǫ/ λǫ 1 ǫ. Proof: For every decomposition ρ ǫ = i p iσ i, we have ψ ρ ǫ ψ = i p i ψ σ i ψ = 1 3ǫ/4. Then there exists i such that ψ σ i ψ 1 3ǫ/4. Therefore, by definition µ σ i λǫ. This means that for any such decomposition we have max i µσ i λǫ, and then µ entρ ǫ λǫ. For the other direction let σ be a state with ψ σ ψ 1 3ǫ/4 and µσ = λǫ. Let us define τ := U U σ U U du, 8 where du denotes the Haar measure on the unitary group. τ is an isotropic state [16], i.e., for every unitary U we have U U τ U U = τ. Any isotropic state is of the form τ = 1 δ ψ ψ +δi /4 = ρ δ. On the other hand, since U U ψ = ψ for every unitary U, we have 1 3δ/4 = ψ τ ψ = ψ σ ψ 1 3ǫ/4. Therefore δ ǫ. Now observe that 8 is already a decomposition of τ = ρ δ. Moreover, maximal correlation does not change under local unitaries. Therefore, we have µ ent ρ δ µσ = λǫ. To finish the prove we only need to notice that δ ǫ and then µ ent ρ δ µ entρ ǫ. The latter inequality holds because ρ ǫ = 1 ǫ 1 δ ρδ + 1+ǫ δ I/4, 1 δ and that maximal entanglement is quasi-convex. λǫ = 0 for ǫ /3 is implied by the partial transpose test as discussed above. The lower bound λǫ 1 3ǫ/ is an immediate consequence of Lemma 1. The upper bound λǫ 1 ǫ is because ǫ ǫ µ ent ρ µ ρ. IV. CONCLUDING REMARKS In this paper we defined a measure of entanglementµ ent for quantum states that is faithful, monotone under local quantum operations, and gives the same number when computed on tensor powers of a bipartite state. The latter property, in particular, implies that maximal entanglement is neither superadditive nor monogamous. As we saw in Theorem 7 a measure of entanglement with the above properties is either not monotone under classical communication or achieves its maximum value on all distillable states.

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