On correlations and mutual entropy in quantum composed systems
Yuji Hirota, Dariusz Chruściński, Takashi Matsuoka, Masanori Ohya
aa r X i v : . [ qu a n t - ph ] S e p On correlations and mutual entropyin quantum composite systems
Yuji HIROTADepartment of Mathematics, Tokyo University of Science, Tokyo, 162-8601, JapanDariusz CHRU ´SCI ´NSKIInstitute of Physics, Nicolaus Copernicus University, Toru´n, 87-100, PolandTakashi MATSUOKADepartment of Business Administration and Information,Tokyo University of Science, Suwa, Nagano, 391-0292, JapanandMasanori OHYADepartment of Information Science, Tokyo University of Science, Chiba, 278-8510, Japan
Abstract
We study the correlations of classical and quantum systems from the information theoretical points ofview. We analyze a simple measure of correlations based on entropy (such measure was already investi-gated as the degree of entanglement by Belavkin, Matsuoka and Ohya). Contrary to naive expectation, itis shown that separable state might possesses stronger correlation than an entangled state.
Correlations play a key role both in classical and quantum physics. In particular the study of correlationsis crucial in many-body physics and classical and quantum statistical physics. Recently, it turned out thatcorrelations play prominent role in quantum information theory and many modern applications of quantumtechnologies and there are dozens of papers dealing with this problem (for the recent review see e.g. [27]).The aim of this paper is to analyze classical and quantum correlations encoded in the bi-partite quan-tum states. Beside quantum entanglement we analyze a new measure – so called D -correlations – and thequantum discord. We propose to compare correlations of di ff erent bi-partite states with the same reducesstates, i.e. locally they contain the same information. It is shown that surprisingly a separable state maybe more correlated that an entangled one. Analyzing simple examples of Bell diagonal states we illus-trate the behavior of various measures of correlations. We also provide an introduction to bi-partite statesand entanglement mappings introduced by Belavkin and Ohya and recall basic notions from classical andquantum information theory. An entanglement mapping encodes the entire information about a bi-partitequantum state and hence it provides an interesting way to deal with entanglement theory. Interestingly, itmay be applied in infinite-dimensional case and in the abstract C ∗ -algebraic settings. Therefore, in a sense,it provides a universal tool in entanglement theory.The paper is organized as follows: in the next section we recall basic facts from the theory of compositequantum systems and introduce the notion of entanglement mappings. Moreover, we recall an interestingconstruction of quantum conditional probability operators. Section 3 recall classical and quantum entropicquantities and collects basic facts from classical and quantum information theory. In particular it containsthe new measure of correlation called D -correlation. Section 4 recalls the notion of quantum discord whichwas intensively analyzed recently in the literature. In section 5 we recall the notion of a circulant state andprovide several examples of states for which one is able to compute various measures of correlations. Finalconclusions are collected in the last section.Throughout the paper, we use standard notation: H , K for complex separable Hilbert spaces and denotethe set of the bounded operators and the set of all states on H by B ( H ) and S ( H ), respectively. In the1 -dimensional Hilbert space, the standard basis is denoted by { e , e , · · · , e d − } and the inner product isdenoted by h· , ·i . We write e i j for | e i ih e j | . Given any state θ on the tensor product Hilbert space H ⊗ K , wedenote by Tr K θ the partial trace of θ with respect to K . Consider a quantum system living in the Hilbert space H . In this paper we consider only finite dimen-sional case. However, as we shall see several results may be nicely generalized to the infinite-dimensionalsetting. Denote by T ( H ) a set of trace class operators in H , meaning that ρ ∈ T ( H ) if ρ ≥ ρ < ∞ ,which is always true in finite-dimensional case. Finally, let S ( H ) = { ρ ∈ T ( H ) | Tr ρ = } , Consider now a composite system living in
H ⊗ K and denote by S SEP ⊂ S ( H ⊗ K ) a convex subset ofseparable states in
H ⊗ K . Recall that ρ ∈ S ( H ⊗ K ) is separable if ρ = P α p α η α ⊗ σ α , where η α ∈ S ( H )and σ α ∈ S ( K ), and p α denotes probability distribution: p α ≥ P α p α =
1. A state ρ ∈ S ( H ⊗ K ) iscalled positive partial transpose (PPT) if its partial transpose satisfies (id H ⊗ τ ) ρ ≥
0, where id H denotes anidentity map in B ( H ). It means that ρ is PPT if (id H ⊗ τ ) ρ ∈ S ( H ⊗ K ). Denote by S PPT a convex subsetof PPT states. It is well known [41] that S ( H ⊗ K ) ⊃ S PPT ⊃ S SEP . In general, the PPT condition is notsu ffi cient for separability.Interestingly, due to the well known duality between states living in H ⊗ K and linear maps B ( K ) → B ( H ),one may translate the above setting in terms of linear maps. Let us recall basic facts concerning completelypositive maps [40]. A linear map χ : B ( K ) → B ( H ) is said to be completely positive (CP) if, for any n ∈ N ,the map χ n : M n ( C ) ⊗ B ( K ) −→ M n ( C ) ⊗ B ( H ) , ( a i , j ) i , j (cid:0) χ ( a i , j ) (cid:1) i , j (2.1)is positive, where B ( H ) denotes bounded operators in H and M n ( C ) stands for n × n matrices with entriesin C . A linear map χ : B ( K ) → B ( H ) is said to be completely copositive (CCP) if composed withtransposition τ , i.e. τ ◦ χ , is CP.Consider now a state θ ∈ S ( H ⊗ K ) and let φ : B ( K ) → B ( H ) be a linear map defined by φ ( b ) : = Tr K [(1 H ⊗ b ) θ ] , for any b ∈ B ( K ). The dual map φ ∗ reads φ ∗ ( a ) = Tr H [( a ⊗ K ) θ ] , for any b ∈ B ( H ). It should be stressed that the above construction is perfectly well defined also in theinfinite-dimensional case if wew assume that θ is a normal state, that is, it is represented by the densityoperator. Note, that a state θ and the linear map φ give rise a linear functional ω : B ( H ⊗ K ) → C ω ( a ⊗ b ) : = Tr( a ⊗ b ) θ, (2.2)for any a ∈ B ( H ) , b ∈ B ( K ). This formula may be equivalently rewritten as follows ω ( a ⊗ b ) = Tr H a φ ( b ) = Tr K φ ∗ ( a ) b . (2.3)It is clear that the marginal states readTr K θ = φ (1 K ) ∈ B ( H ) , Tr H θ = φ ∗ (1 H ) ∈ B ( K ) . (2.4)Belavkin and Ohya observed [11, 12] that if θ ∈ S ( H ⊗ K ), then both φ and its dual φ ∗ are CCP. We denoteby B ( H ) the dual space to the algebra B ( H ). Definition 2.1
A CCP map φ : B ( K ) → B ( H ) normalized as Tr H φ (1 K ) = is called the entanglementmap from ρ : = φ ∗ (1 H ) ∈ B ( K ) to σ : = φ (1 K ) ∈ B ( H ) . A density operator θ φ corresponding to the entanglement map φ with its marginals φ ∗ (1 H ) and φ (1 K )can be represented as follows: let ψ + K denotes a maximally entangled state in K ⊗ K . Then θ φ : = ( φ ⊗ τ ) P + K , (2.5)2ith P + K = d K | ψ + K ih ψ + K | , where d K = dim K . If { e k } stands for an orthonormal basis in K , then P + K = d K X i , j = e i j ⊗ e i j , (2.6)with e i j : = | e i ih e j | , and hence θ φ = d K X i , j = φ ( e ji ) ⊗ e i j . (2.7)The map assigning θ φ to φ is usually called a Choi-Jamiołkowski isomorphism. It should be stressed that θ φ does not depend upon the choice of { e k } . Lemma 2.2
A linear map φ : B ( K ) → B ( H ) is CCP if and only if θ φ ≥ . Clearly, φ is CP if and only if φ ◦ τ is CCP. Due to Lemma 2.2, we have the following criterion.
Theorem 2.3 [29, 32] A state θ φ is a PPT state if and only if its entanglement map φ is CP. Recently, Kossakowski et al.[5] proposed the following construction: for θ ∈ S ( H ⊗ K ) one defines thebounded operator π θ : = (cid:0) ρ − ⊗ K (cid:1) θ (cid:0) ρ − ⊗ K (cid:1) , (2.8)where ρ : = Tr K θ . It is verified that π θ satisfies π θ ≥ , (2.9)Tr K π θ = H ∈ B ( H ) . (2.10)In what follows we assume that ρ is a faithful state, i.e. ρ >
0. It follows from ( 2.9) and (2.10) that theoperator π θ is the quantum analogue of a classical conditional probability. Indeed, if B ( H ⊗ K ) is replacedby commutative algebra, then π θ coincides with a classical conditional probability. Definition 2.4
An operator π ∈ B ( H ⊗ K ) is called the quantum conditional probability operator (QCPO,for short) if π satisfies condition (2.9) and (2.10). It is easy to verify[5] that for any CP unital map ϕ : B ( K ) → B ( H ) and an orthonormal basis in K thefollowing operator π ϕ = d K X k , l = ϕ ( e kl ) ⊗ e kl , (2.11)defines QCPO. From Lemma 2.2 and unitality of ϕ , it follows that π ϕ satisfies conditions (2.9) and (2.10).For a given π ϕ and any faithful marginal state ρ ∈ S ( H ), one can construct a state θ of the composite system θ ϕ = d K X k , l = ρ ϕ ( e kl ) ρ ⊗ e kl . (2.12)It is clear that θ ϕ is a PPT state if and only if the map ϕ is a CCP. There exists a simple relation betweenthe density operator θ φ in (2.7) and the QCPO π ϕ in (2.11) due to the following decomposition of theentanglement map φ . Lemma 2.5 [13] Every entanglement map φ with φ (1 K ) = ρ has a decomposition φ ( · ) = ρ ϕ ◦ τ ( · ) ρ , (2.13) where ϕ is a CP unital map to be found as a unique solution to ϕ ( · ) = ρ − φ ◦ τ ( · ) ρ − . (2.14) Theorem 2.6 [20] If a composite state θ φ given by (2.7) has a faithful marginal state ρ = φ (1 K ) , then θ φ isrepresented by θ φ = (cid:0) ρ ⊗ K (cid:1) π φ (cid:0) ρ ⊗ K (cid:1) , (2.15) where π φ = P k , l ρ − φ ( e kl ) ρ − ⊗ e kl . Classical and quantum information
In classical description of a physical composite system its correlation can be represented by a jointprobability measure or a conditional probability measure. In classical information theory we have propercriteria to estimate such correlation, which are so-called the mutual entropy and the conditional entropygiven by Shannon [42]. Here we review Shannon’s entropies briefly.Let X = { x i } ni = and Y = { y j } mj = be random variables with probability distributions p i and q j , respectively,and let p i | j denotes conditional probability P ( X = x i | Y = y j ). The joint probability r i j = P ( X = x i , Y = y j )is given by r i j = p i | j q j . (3.1)Let us recall definitions of mutual entropy I ( X : Y ) and conditional entropies S ( X | Y ) , S ( Y | X ): I ( X : Y ) = X i , j r i j log r i j p i q j , and S ( X | Y ) = − X j q j X i p i | j log p i | j , S ( Y | X ) = − X i p i X j p j | i log p j | i . Using (3.1), we can easily check that the following relations I ( X : Y ) = S ( X ) + S ( Y ) − S ( XY ) , (3.2)and S ( X | Y ) = S ( XY ) − S ( Y ) = S ( X ) − I ( X : Y ) , (3.3) S ( Y | X ) = S ( XY ) − S ( X ) = S ( Y ) − I ( X : Y ) , (3.4)where S ( X ) = − P i p i log p i , and S ( XY ) = − P i j r i j log r i j . Note, that p i | j gives rise to a stochastic matrix T i j : = p i | j and hence it defines a classical channel p i = X j T i j q j . (3.5)Note, that data provided by r i j are the same as those provided by T i j and p j . Hence one may instead of I ( X : Y ) use the following notation I ( P , T ), where P represent an input state and T the classical channel.One interprets I ( P , T ) as a information transmitted via a channel T . The fundamental Shannon inequality0 ≤ I ( P ; T ) ≤ min (cid:8) S ( X ) , S ( Y ) (cid:9) , (3.6)gives the obvious bounds upon the transmitted information.Now, we extend the classical mutual entropy to the quantum system using the Umegaki relative entropy.[43]Let θ ∈ S ( H ⊗ K ) with marginal states ρ ∈ S ( H ) and σ ∈ S ( K ). One defines quantum mutual entropy as arelative entropy between θ and the product of marginals ρ ⊗ σ : I ( θ ) = S ( θ || ρ ⊗ σ ) = Tr { θ (cid:0) log θ − log[ ρ ⊗ σ ] (cid:1) } . (3.7)As in the classical case one shows that I ( θ ) = S ( ρ ) + S ( σ ) − S ( θ ) . (3.8)Introducing quantum conditional entropy S θ ( ρ | σ ) : = S ( θ ) − S ( σ ) , (3.9)one finds I ( θ ) = S ( ρ ) − S θ ( ρ | σ ) , (3.10)or, equivalently I ( θ ) = S ( σ ) − S θ ( σ | ρ ) . (3.11)4 efinition 3.1 [11, 12, 14, 22] For any entanglement map φ : B ( K ) → B ( H ) with ρ = φ (1 K ) and σ = φ ∗ (1 H ) , the quantum mutual entropy I φ ( ρ : σ ) is defined byI φ ( ρ : σ ) : = S ( θ φ || ρ ⊗ σ ) = Tr { θ φ (cid:0) log θ φ − log[ ρ ⊗ σ ] (cid:1) } , (3.12) where S ( · || · ) is the Umegaki relative entropy. One easily finds I φ ( ρ : σ ) = S ( ρ ) + S ( σ ) − S ( θ φ ) . (3.13)The above relation (3.13) is a quantum analog of (3.2). One defines the quantum conditional entropies asgeneralizations of (3.3), (3.4) [11, 12, 14, 24]: S φ ( σ | ρ ) : = S ( σ ) − I φ ( ρ : σ ) = S ( θ φ ) − S ( ρ ) . (3.14)It is usually assumed that I φ ( ρ : σ ) measures all correlations encoded into the bipartite state θ φ withmarginals ρ and σ . Example 3.2 (Product state)
For the entanglement map φ ( b ) : = ρ Tr K ( σ b ) , one finds θ φ = ρ ⊗ σ , and henceI φ ( ρ : σ ) = , S θ (cid:0) σ | ρ (cid:1) = S ( σ ) , S θ (cid:0) ρ | σ (cid:1) = S ( ρ ) , (3.15) which recover well known relations for a product state ρ ⊗ σ . Example 3.3 (Pure entangled state)
Let { λ i } be the sequence of complex numbers satisfying P i | λ i | = .For entanglement mappings φ ( b ) = r X i , j = λ i λ j e i j h f j , b f i i , where { e k } and { f l } are orthonormal basis in H and K , respectively, the state θ φ can be written in thefollowing form θ φ = r X i , j = λ i λ j e i j ⊗ f i j = (cid:12)(cid:12)(cid:12) Ψ ih Ψ (cid:12)(cid:12)(cid:12) , where (cid:12)(cid:12)(cid:12) Ψ (cid:11) = r X i = λ i e i ⊗ f i ∈ H ⊗ K . Note, that r ≤ min { d H , d K } , equals to the Schmidt rank of Ψ ∈ H ⊗ K . One finds for the reduced states ρ = φ (1 K ) = r X i = | λ i | e ii , σ = φ ∗ (1 H ) = r X i = | λ i | f ii , and hence I φ ( ρ : σ ) = S ( ρ ) + S ( σ ) − S ( θ ) = S ( ρ ) > min (cid:8) S ( ρ ) , S ( σ ) (cid:9) , (3.16) together with S θ ( σ | ρ ) = S θ ( ρ | σ ) = − S ( ρ ) < , (3.17) where S ( ρ ) = S ( σ ) = − P ri = | λ i | log | λ i | .
5s is mentioned in Section 2, the classical mutual entropy always satisfies the Shannon’s fundamentalinequality, i.e. it is always smaller than its marginal entropies, and the conditional entropy is always positive.Note that separable state has the same property. It is no longer true for pure entangled states.Now we introduce another measure for correlation of composite states.[11, 12, 20, 34]
Definition 3.4
For the entanglement map φ : B ( K ) → B ( H ) , we define the D-correlation D ( θ ) of θ asD ( θ ) : = − { S θ ( σ | ρ ) + S θ ( ρ | σ ) } =
12 ( S ( ρ ) + S ( σ )) − S ( θ ) . (3.18)Note that the D -correlation with the opposite convention − D ( θ ) is called the degree of entanglement.[11,12, 20, 34] One proves the following: Proposition 3.5 [2, 34] If θ φ is a pure state, then the following statements hold:1. θ is entangled state if and only if D ( θ ) > .2. θ is separable state if and only if D ( θ ) = . It is well-known that if θ is a PPT state, then S ( θ ) − S ( ρ ) ≥ , S ( θ ) − S ( σ ) ≥ , (3.19)where ρ and σ are the marginal states of θ .[44] Proposition 3.6 If θ is a PPT state, then D ( θ ) ≤ . (3.20)Suppose now that we have two entanglement mappings φ k : B ( K ) → B ( H ) , ( k = ,
2) such that φ (1 K ) = φ (1 K ) and φ ∗ (1 H ) = φ ∗ (1 H ). Let θ , θ ∈ S ( H ⊗ K ) be the corresponding states. We proposethe following:
Definition 3.7 θ is said to have stronger D-correlations than θ ifD ( θ ) > D ( θ ) . (3.21)Several measures of correlation based on entropic quantities were already discussed by Cerf and Adami[14],Horodecki[24], Henderson and Vedral[23], Groisman et al.[22]. Let us briefly recall the definition of quantum discord [39, 23]. Recall, that mutual information may berewritten as follows I ( θ ) = S ( σ ) − S θ ( σ | ρ ) . (4.1)An alternative way to compute the conditional entropy S θ ( σ | ρ ) goes as follows: one introduces a measure-ment on H -party defined by the collection of one-dimensional projectors { Π k } in H satisfying Π +Π + . . . = H . The label ‘ k ’ distinguishes di ff erent outcomes of this measurement. The state after the measurementwhen the outcome corresponding to Π k has been detected is given by θ K| k = p k ( Π k ⊗ K ) θ ( Π k ⊗ K ) , (4.2)where p k is a probability that H -party observes k th result, i.e. p k = Tr( Π k ρ ), and θ K| k is the (collapsed)state in H ⊗ K , after H -party has observed k th result in her measurement. The entropies S ( θ K| k ) weightedby probabilities p k yield the conditional entropy of part K given the complete measurement { Π k } on the part H S ( θ |{ Π k } ) = X k p k S ( θ K| k ) . (4.3)Finally, let I ( θ |{ Π k } ) = S ( σ ) − S ( θ |{ Π k } ) , (4.4)6e the corresponding measurement induced mutual information. The quantity C H ( θ ) = sup { Π k } I ( θ |{ Π k } ) , (4.5)is interpreted [39, 23] as a measure of classical correlations. Now, these two quantities – I ( θ ) and C H ( θ ) –may di ff er and the di ff erence D H ( θ ) = I ( θ ) − C H ( θ ) (4.6)is called a quantum discord.Evidently, the above definition is not symmetric with respect to parties H and K . However, one caneasily swap the role of H and K to get D K ( θ ) = I ( θ ) − C K ( θ ) , (4.7)where C K ( θ ) = sup { e Π α } I ( θ |{ e Π α } ) , (4.8)and e Π α is a collection of one-dimensional projectors in K satisfying e Π + e Π + . . . = K . For a general mixedstate D H ( θ ) , D K ( θ ). However, it turns out that D H ( θ ) , D K ( θ ) ≥
0. Moreover, on pure states, quantumdiscord coincides with the von Neumann entropy of entanglement S ( ρ ) = S ( σ ). States with zero quantumdiscord – so called classical-quantum states – represent essentially a classical probability distribution p k embedded in a quantum system. One shows that D H ( θ ) = | k i in H such that θ = X k p k | k ih k | ⊗ σ k , (4.9)where σ k are density matrices in K . Similarly, D K ( θ ) = | α i in K such that θ = X α q α ρ α ⊗ | α ih α | , (4.10)where ρ α are density matrices in H . It is clear that if D H ( θ ) = D K ( θ ) =
0, then θ is diagonal in the productbasis | k i ⊗ | α i and hence θ = X k ,α λ k α | k ih k | ⊗ | α ih α | , (4.11)is fully encoded by the classical joint probability distribution λ k α .Finally, let us introduce a symmetrized quantum discord D H : K ( θ ) : = h D H ( θ ) + D K ( θ ) i . (4.12)Let us observe that there is an intriguing relation between (4.12) and (3.18). One has D ( θ ) = I ( θ ) −
12 [ S ( ρ ) + S ( σ )] , (4.13)whereas D H : K ( θ ) = I ( θ ) − C H : K ( θ ) . (4.14)Note, that D H : K ( θ ) ≥ D ( θ ) can be negative (for PPT states). It is assumed that D H : K ( θ ) measuresperfectly quantum correlations encoded into θ . Example 4.1 (Separable correlated state)
For the entanglement map given by φ ( b ) = X i λ i ρ i Tr σ i b , φ ∗ ( a ) = X i λ i σ i Tr ρ i a , (cid:18) X i λ i = , λ i ≥ ∀ i (cid:19) , the corresponding state θ can be written in the form θ = X i λ i ρ i ⊗ σ i , (4.15) with ρ = φ (1 K ) = P i λ i ρ i and σ = φ ∗ (1 H ) = P i λ i σ i . Then, we have the following inequalities.[3, 11, 12] ≤ I ( θ ) ≤ min (cid:8) S ( ρ ) , S ( σ ) (cid:9) , (4.16) S θ ( σ | ρ ) ≥ , S θ ( ρ | σ ) ≥ . (4.17)7 xample 4.2 (Separable perfectly correlated state) Let { e i } i and { f j } j be the complete orthonormal sys-tems in H and K , respectively. For the entanglement map given by φ ( b ) = X i λ i | e i ih e i |h f i , b f i i , φ ∗ ( a ) = X λ i | f i ih f i |h e i , ae i i , the corresponding state θ can be written in the form θ = X λ i | e i ih e i | ⊗ | f i ih f i | , with ρ = φ (1 K ) = P λ i | e i ih e i | , σ = φ ∗ (1 H ) = P i λ i | f i ih f i | . It is clear that D H : K ( θ ) = . Moreover, oneobtains I ( θ ) = S ( ρ ) + S ( σ ) − S ( θ φ ) = S ( ρ ) , (4.18) S θ ( σ | ρ ) = S θ ( ρ | σ ) = , (4.19) where S ( ρ ) = S ( σ ) = S ( θ φ ) = − P λ i log λ i . This correlation corresponds to a perfect correlation in theclassical scheme. In this section, we analyze correlations encoded into the special family of so called circulant states . We start this section by recalling the definition of a circulant state introduced in [17] (see also [18]).Consider the finite dimensional Hilbert space C d with the standard basis { e , e , · · · , e d − } . Let Σ be thesubspace of C d ⊗ C d generated by e i ⊗ e i ( i = , , · · · , d −
1) : Σ = span { e ⊗ e , e ⊗ e , · · · , e d − ⊗ e d − } . (5.1)Define a shift operator S α : C d → C d by S α e k = e k + α , mod d and let Σ α : = (1 d ⊗ S α ) Σ . (5.2)It turns out that Σ α and Σ β ( α , β ) are mutually orthogonal and one has the following direct sum decompo-sition C d ⊗ C d (cid:27) Σ ⊕ Σ ⊕ · · · ⊕ Σ d − . (5.3)This decomposition is called a circulant decomposition.[17] Let a (0) , a (1) , · · · , a ( d − be positive d × d matrices with entries in C such that ρ α is supported on Σ α . Moreover, lettr( a (0) + · · · + a ( d − ) = . (5.4)Now, for each a ( α ) ∈ M d ( C ) one defines a positive operator in C d ⊗ C d be the following formula ϑ α = d − X i , j = a ( α ) i j e i j ⊗ S α e i j S α † . (5.5)Finally, let us introduce ϑ : = ϑ ⊕ · · · ⊕ ϑ d − . (5.6)One proves[17] that ρ defines a legitimate density operators in C d ⊗ C d . One calls it a circulant state . Forfurther details of circulant states we refer to Refs. [17, 18].Now, let consider a partial transposition of the circulant state. It turns out that ρ τ = (1l ⊗ τ ) ρ is againcirculant but it corresponds to another cyclic decomposition of the original Hilbert space C d ⊗ C d . Let us8ntroduce the following permutation π from the symmetric group S d : it permutes elements { , , . . . , d − } as follows π (0) = , π ( i ) = d − i , i = , , . . . , d − . (5.7)We use π to introduce e Σ = span (cid:8) e ⊗ e π (0) , e ⊗ e π (1) , . . . , e d − ⊗ e π ( d − (cid:9) , (5.8)and e Σ α = (1l ⊗ S α ) e Σ . (5.9)It is clear that e Σ α and e Σ β are mutually orthogonal (for α , β ). Moreover, e Σ ⊕ . . . ⊕ e Σ d − = C d ⊗ C d , (5.10)and hence it defines another circulant decomposition. Now, the partially transformed state ϑ τ has again acirculant structure but with respect to the new decomposition (5.10): ϑ τ = e ϑ (0) + · · · + e ϑ ( d − , (5.11)where e ϑ ( α ) = d − X i , j = e a ( α ) i j e i j ⊗ S α e π ( i ) π ( j ) S † α , α = , . . . , d − , (5.12)and the new d × d matrices [ e a ( α ) i j ] are given by the following formulae: e a ( α ) = d − X β = a ( α + β ) ◦ ( Π S β ) , mod d , (5.13)where “ ◦ ” denotes the Hadamard product, and Π being a d × d permutation matrix corresponding to π , i.e. Π i j : = δ i ,π ( j ) . It is therefore clear that our original circulant state is PPT i ff all d matrices e a ( α ) satisfy e a ( α ) ≥ , α = , . . . , d − . (5.14) The most important example of circulant states is provided by Bell diagonal states [6, 7, 8] defined by ρ = d − X m , n = p mn P mn , (5.15)where p mn ≥ P m , n p mn = P mn = ( I ⊗ U mn ) P + d ( I ⊗ U † mn ) , (5.16)with U mn being the collection of d unitary matrices defined as follows U mn e k = λ mk S n e k = λ mk e k + n , (5.17)with λ = e π i / d . (5.18)The matrices U mn define an orthonormal basis in the space M d ( C ) of complex d × d matrices. One easilyshows Tr( U mn U † rs ) = d δ mr δ ns . (5.19)Some authors call U mn generalized spin matrices since for d = U = I , U = σ , U = i σ , U = σ . (5.20) A Hadamard (or Schur) product of two n × n matrices A = [ A ij ] and B = [ B ij ] is defined by( A ◦ B ) ij = A ij B ij . C d ⊗ C d . Indeed, maximally entangledprojectors P mn are supported on Σ n , that is, Π n = P n + . . . + P d − , n , (5.21)defines a projector onto Σ n , i.e. Σ n = Π n ( C d ⊗ C d ) . (5.22)One easily shows that the corresponding matrices a ( n ) are given by a ( n ) = HD ( n ) H ∗ , (5.23)where H is a unitary d × d matrix defined by H kl : = √ d λ kl , (5.24)and D ( n ) is a collection of diagonal matrices defined by D ( n ) kl : = p kn δ kl . (5.25)One has a ( n ) kl = d d − X m = p mn λ m ( k − l ) , (5.26)and hence it defines a circulant matrix a ( n ) kl = f ( n ) k − l , (5.27)where the vector f ( n ) m is the inverse of the discrete Fourier transform of p mn ( n is fixed). Let H = K = C . For any α ∈ [0 , θ ( α ) = P + + α Π + − α Π . (5.28)The eigenvalues of θ ( α ) are calculated as 0 , , × α and 3 × − α and hence one obtains for the D -correlations D (cid:0) θ ( α ) (cid:1) = log 3 +
27 log 27 + α α + − α − α . (5.29) Theorem 5.1 [26] The family θ ( α ) satisfies:1. θ ( α ) is PPT if and only α ∈ [1 , θ ( α ) is separable if and only if α ∈ [2 , ;3. θ ( α ) is both entangled and PPT if and only if α ∈ [1 , ∪ (3 , ;4. θ ( α ) is NPT if and only if α ∈ [0 , ∪ (4 , . Due to this Theorem, one can find that the D ( θ ( α )) does admit a natural order. That is, the D -correlationfor any entangled state is always stronger than D -correlation for an arbitrary separable state. Similarly, oneobserves that D -correlation for any NPT state is always stronger than D -correlation for an arbitrary PPTstate. The graph of D (cid:0) θ ( α ) (cid:1) is shown in Fig. 2. Actually, one finds that the minimal value of D -correlationscorresponds to α = .
5, that is, it lies in the middle of the separable region.On the other hand, we can also compute the symmetrized discord D C ; C ( θ ( α )) and have obtainedFig. 2. It is easy to find that the graph is symmetric with respect to α = .
5. As in Fig. 2, the value of thesymmetrized discord satisfies the following inequality;0 < D C ; C ( θ ( α )) ≤ D C ; C ( θ ( β )) ≤ D C ; C ( θ ( γ )) , - - - - Figure 1: Left — the graph of D ( θ ( x )) with x ∈ [0 , D corresponds to x = . D C ; C ( θ ( α )).where α ∈ [2 , , β ∈ [1 , ∪ [3 ,
4] and γ ∈ [0 , ∪ [4 , θ ( α ) has the quantum correlation even in separable states corresponding to α ∈ [2 ,
3] inthe sense of discord. We know that the above two types of criteria give the similar order of correlation.Notice that D ( θ ( α )) is always negative even in NPT sates and the positivity of D -correlation representsa true quantum property (see Example 3.3 and Proposition 3.5). In this sense the quantum correlation of θ ( α ) is not so strong.This family may be generalized to C d ⊗ C d as follows: consider the following family of circulat 2-quditstates θ ( α ) = d − X i = λ i Π i + λ d P + d , (5.30)with λ n ≥
0, and λ + . . . + λ d − + λ d =
1. Let us take the following special case corresponding to λ = αℓ , λ d − = ( d − + − αℓ , λ d = d − ℓ . (5.31)and λ = . . . = λ d − = λ d , with ℓ = ( d − d − + . (5.32)One may prove the following[21] Theorem 5.2
The family θ ( α ) satisfies:1. θ ( α ) is PPT if and only α ∈ [1 , ( d − ] θ ( α ) is separable if and only if α ∈ [ d − , ( d − d − + ;3. θ ( α ) is both entangled and PPT if and only if α ∈ [1 , d − ∪ (( d − d − + , ( d − ] ;4. θ ( α ) is NPT if and only if α ∈ [0 , ∪ (( d − , ( d − + . For example if d = D ( θ ( α )) (see Fig. 4) Again, one finds that - - - - Figure 2: The graph of D ( θ ( x )) with x ∈ [0 , D corresponds to x = D ( θ ( α )) does admit a natural order. That is, the D -correlation for any entangled state is always strongerthan D -correlation for an arbitrary separable state. Similarly, one observes that D -correlation for any NPTstate is always stronger than D -correlation for an arbitrary PPT state.11 .4 Example: a family of Bell diagonal states Consider the following class of Bell-diagonal states in C ⊗ C : θ ( ε ) = Λ (3 P + + ε Π + ε − Π ) , (5.33)with Λ = + ε + ε − . One easily finds for its D -correlations D (cid:0) θ ( ε ) (cid:1) = Λ log 1 Λ + ε − log ε − Λ + ε log ε Λ + log 3 ! . (5.34)The following theorem gives us a useful characterization of θ ( ε ) [30]. Theorem 5.3
The states of θ ( ε ) are classified by ε as follows:1. θ ( ε ) is separable if ε = ;2. θ ( ε ) is both PPT and entangled for ε , . The graph of D (cid:0) θ ( ε ) (cid:1) is shown in Fig. 3. D (cid:0) θ ( ε ) (cid:1) is rapidly decreasing with ε approaching 1 from 0 andincreases when ε is over 1. That is, D (cid:0) θ ( ε ) (cid:1) takes the minimal value at ε = D (cid:0) θ (1) (cid:1) = − log 3 ≈ − . θ ( α ), the D -correlation D (cid:0) θ ( ε ) (cid:1) for an entangled state isalways stronger than the one for a separable state. As ε → ∞ , θ ( ε ) converges to a separable perfectlycorrelated state which can be recognized as a “classical state”lim ε → θ ( ε ) = (cid:16) e ⊗ e + e ⊗ e + e ⊗ e (cid:17) = Π , (5.35)lim ε →∞ θ ( ε ) = (cid:16) e ⊗ e + e ⊗ e + e ⊗ e (cid:17) = Π , (5.36)and for every ε > D (cid:0) θ ( ε ) (cid:1) < = lim ε → D (cid:0) θ ( ε ) (cid:1) = lim ε →∞ D (cid:0) θ ( ε ) (cid:1) . (5.37)It shows that a correlation of a PPT entangled state θ ( ε ,
1) is weaker than that of the (classical) separableperfectly correlated states in the sense of (3.21).Now, since θ ( α ) and θ ( ε ) have common marginal states, we can compare the order of quantum corre-lations for them. One has, for example, D (cid:0) θ (1) (cid:1) ≈ − . > − . ≈ D (cid:0) θ (3 . (cid:1) . (5.38)Accordingly Theorem 5.1 and 5.3, however, θ (1) is separable while θ (3 .
1) is entangled state. Incidentally,this means that the correlation for the separable state θ (1) is stronger than the entangled state θ (3 .
1) in thesense of (3.21). - - - - Figure 3: Left — the graph of D ( θ ( x )). Note that D is minimal for x = D C : C ( θ ( ε ) for ε ∈ (0 , D C : C ( θ ( ε )) = D C : C ( θ ( ε − )).On the other hand one finds the following plot of the quantum discord Fig. 3.It is clear that lim ε → D C : C ( θ ( ε )) = lim ε →∞ D C : C ( θ ( ε )) = , (5.39)since both Π and Π are perfectly classical states. Note, that D C : C ( θ ( ε = > θ ( ε =
1) does contain quantum correlations.12
Conclusions
We provided several examples of bi-partite quantum states and computed two types of correlations forthem. It turned out that the correlation for a separable state can be stronger than the one for an entangledstate in the sense of (3.21). This observation is inconsistent with the conventional understanding of quantumentanglement. However, we also showed that the discord of such separable states might strictly positive.This means that these states have a non-classical correlation. From this point of view, it is no longer unusualthat the correlation for a separable state is stronger than the one for an entangled state.
Acknowledgments
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