aa r X i v : . [ qu a n t - ph ] M a y Non-commutativity measure of quantum discord
Yu Guo a School of Mathematics and Computer Science, Shanxi Datong University, Datong, Shanxi 037009, China
Quantum discord is a manifestation of quantum correlations due to non-commutativity ratherthan entanglement. Two measures of quantum discord by the amount of non-commutativity viathe trace norm and the Hilbert-Schmidt norm respectively are proposed in this paper. These twomeasures can be calculated easily for any state with arbitrary dimension. It is shown by severalexamples that these measures can reflect the amount of the original quantum discord. a Correspondence to [email protected]
Introduction
The characterization of quantum correlations in composite quantum states is of great importance in quantum informa-tion theory [1–6]. It has been shown that there are quantum correlations that may arise without entanglement, suchas quantum discord (QD) [4], measurement-induced nonlocality (MIN)[6], quantum deficit [7], quantum correlationinduced by unbiased bases [8, 9] and quantum correlation derived from the distance between the reduced states [10],etc. Among them, quantum discord has aroused great interest in the past decade [11–30]. It is more robust againstthe effects of decoherence [13] and can be a resource in quantum computation [31, 32], quantum key distribution [33]remote state preparation [34, 35] and quantum cryptography [36].Quantum discord is initially introduced by Ollivier and Zurek [4] and by Henderson and Vedral [5]. The idea is tomeasure the discrepancy between two natural yet different quantum analogs of the classical mutual information. Fora state ρ of a bipartite system A+B described by Hilbert space H a ⊗ H b , the quantum discord of ρ (up to part B) isdefined by D ( ρ ) := min Π b { I ( ρ ) − I ( ρ | Π b ) } , (1)where, the minimum is taken over all local von Neumann measurements Π b , I ( ρ ) := S ( ρ a )+ S ( ρ b ) − S ( ρ ) is interpretedas the quantum mutual information, S ( ρ ) := − Tr( ρ log ρ ) is the von Neumann entropy, I ( ρ | Π b ) } := S ( ρ a ) − S ( ρ | Π b ), S ( ρ | Π b ) := P k p k S ( ρ k ), and ρ k = p k ( I a ⊗ Π bk ) ρ ( I a ⊗ Π bk ) with p k = Tr[( I a ⊗ Π bk ) ρ ( I a ⊗ Π bk )], k = 1, 2, . . . ,dim H b . Calculation of quantum discord given by Eq. (1) in general is NP-complete since it requires an optimizationprocedure over the set of all measurements on subsystem B [37]. Analytical expressions are known only for certainclasses of states [15, 16, 20, 38, 40–46]. Consequently, different versions (or measures) of quantum discord havebeen proposed [19, 24, 25, 39, 47]: the discord-like quantities in [39], the geometric measure [47], the Bures distancemeasure [24] and the trace norm geometric measure [19], etc. Unfortunately, all of theses measures are difficult tocompute since they also need the minimization or maximization scenario.Let {| i a i} be an orthonormal basis of H a . Then any state ρ acting on H a ⊗ H b can be represented by ρ = X i,j E ij ⊗ B ij , (2)where E ij = | i a ih j a | and B ij = Tr a ( | j a ih i a | ⊗ b ρ ). That is, assume that Alice and Bob share a state ρ , if Alice takean ‘operation’ Θ ij : ρ
7→ | j a ih i a | ⊗ b ρ (3)on her part, then Bob obtains the local operator B ij (Note here that, the ‘operation’ Θ ij is not the usual quantumoperation which admits the Kraus sum respresentation). Quantum discord is from non-commutativity: D ( ρ ) = 0 ifand only if B ij s are mutually commuting normal operators [47, 48]. It follows that the non-commutativity of thelocal operators B ij s implies ρ contains quantum discord. The central aim of this article is to show that, for anygiven state written as in Eq. (2), its quantum discord can be measured by the amount of non-commutativity of thelocal operators, B ij s. In the following, we propose our approach: the non-commutativity measures. We present twomeasures: the trace norm measure and the Hilbert-Schmidt norm one. Both of them can be calculated for any statedirectly via the Lie product of the local operators. We then analyze our quantities for the Werner state, the isotropicstate and the Bell-diagonal state in which the original quantum discord have been calculated. By comparing ourquantities with the original one, we find that our quantities can quantify quantum discord roughly for these states. Results
The amount of non-commutativity.
Let X and Y be arbitrarily given operators on some Hilbert space. Then[ X, Y ] = XY − Y X = 0 if and only if k [ X, Y ] k = 0, k·k is any norm defined on the operator space. That is, k [ X, Y ] k 6 = 0implies the non-commutativity of X and Y . In general, k [ X, Y ] k reflects the amount of the non-commutativity of X and Y . Furthermore, for a set of operators Γ = { A i : 1 ≤ i ≤ n } , the total non-commutativity of Γ can be defined by N (Γ) := X i Let ρ = P i,j E ij ⊗ B ij be a state acting on H a ⊗ H b as inEq. (2). We define a measure of QD for ρ by D N ( ρ ) := X i ≤ k,j ≤ l k [ B ij , B kl ] k Tr + X i 0, both D N and D ′ N vanish only for the zero quantum discordstates, i.e., D N ( ρ ) = D ′ N ( ρ ) = 0 iff D ( ρ ) = 0; ii) both D N and D ′ N are invariant under the local unitary operations asthat of the quantum discord, i.e., D N ( ρ ) = D N ( U a ⊗ U b ρU † a ⊗ U † b ) and D ′ N ( ρ ) = D ′ N ( U a ⊗ U b ρU † a ⊗ U † b ) for any unitaryoperator U a/b acting on H a/b (this implies that D N and D ′ N are independent on the choice of the local orthonormalbases : if ρ = P i,j E ij ⊗ B ij with respect to the local orthonormal basis {| i a i| j b i} and ρ = P i,j E ′ ij ⊗ B ′ ij withrespect to another local orthonormal basis {| i ′ a i| j ′ b i} , then E ′ ij = U a E ij U † a and B ′ ij = U b B ij U † b for some local unitaryoperators U a and U b ); iii) D N ( ρ ) ≥ D ′ N ( ρ ) for any ρ . By the definitions, it is clear that both D N and D ′ N can beeasily calculated for any state.Let | ψ i be a pure state with Schmidt decomposition | ψ i = P k λ k | k a i| k b i . Then D N ( | ψ ih ψ | ) = 2 X i,j λ i λ j ( X ( k,l ) ∈ Ω λ k λ l ) , (7) D ′ N ( | ψ ih ψ | ) = 2 X i,j λ i λ j ( X ( k,l ) ∈ Ω ′ λ k λ l ) + √ , (8)where Ω = { ( k, l ) : either i < k ≤ j ≤ l or k = i and l = j if i < j ; i ≤ k < l if i = j } , Ω ′ = { ( k, l ) : i < k ≤ j ≤ l if i < j ; i ≤ k < l if i = j } . Therefore, D N ( | ψ ih ψ | ) = 0 (or D ′ N ( | ψ ih ψ | ) = 0) if and only if | ψ i is separable. Forthe maximally entangled state | Ψ + i = √ d P i | i a i| i b i in a d ⊗ d system, it is straightforward that D N ( | Ψ + ih Ψ + | ) = whenever d = 2, whenever d = 3 and 4 whenever d = 4, D ′ N ( | Ψ + ih Ψ + | ) = 1 + √ whenever d = 2, 2 + √ d = 3 and + √ whenever d = 4. D N and D ′ N reach the maximum values only on the maximally entangled one.It is worth mentioning here that both D N and D ′ N are defined without measurement, so the way we used isfar different from the original quantum discord and other quantum correlations (note that all the measures of thequantum correlations proposed now are defined by some distance between the state and the post state after somemeasurement). In addition, it is clear that D N ( ρ ) and D ′ N ( ρ ) are continuous functions of ρ since both the tracenorm and Hilbert-Schmidt norm are continuous. In [28], a set of criteria for measures of correlations are introduced:(1) necessary conditions ((1-a)-(1-e)), (2) reasonable properties ((2-a)-(2-c)), and (3) debatable criteria ((3-a)-(3-d)). One can easily check that our quantity meets all the necessary conditions as a measure of quantum correlationproposed in [28] (note that the condition (1-d) in [28] is invalid for D N ( ρ ) and D ′ N ( ρ )). The continuity of D N and D ′ N meets the reasonable property (2-a) (note: (2-b) and (2-c) are invalid since these two conditions are associatedwith measurement-induced correlation). (7) and (8) guarantee the debatable property (3-a). (3-c) and (3-d) are notsatisfied as that of the original quantum discord while (3-b) is invalid for D N and D ′ N . That is, all the associatedconditions that satisfied by the original quantum discord are met by our quantities. From this perspective, D N and D ′ N are well-defined measures as that of the original quantum discord. Comparing with the original quantum discord. In what follows, we compare the non-commutativity measures D N and D ′ N with quantum discord D for several classes of well-known states and plot the level surfaces for theBell-diagonal states. These examples will show that D N and D ′ N reflect the amount of quantum discord roughly: D N and D ′ N increase (resp. decrease) if and only if D increase (resp. decrease) for almost all these states (see Figs. 1-3). D N ≥ D and D ′ N ≥ D for almost all these states while there do exist states such that D N < D and D ′ N < D (seeFig. 3 (a-b)). In addition, D N and D ′ N characterize quantum discord in a more large scale than that of D roughly. Forthe two-qubit pure state | ψ i = P k λ k | k a i| k b i , we can also calculate that D N ( | ψ ih ψ | ) > D ( | ψ ih ψ | ) whenever λ > a with a ≈ . D N ( | ψ ih ψ | ) < D ( | ψ ih ψ | ) whenever λ < a and D ′ N ( | ψ ih ψ | ) > D ( | ψ ih ψ | ) whenever λ > b with b ≈ . D ′ N ( | ψ ih ψ | ) < D ( | ψ ih ψ | ) whenever λ < b . Α H a - L D' N D N D 0.2 0.4 0.6 0.8 1.0 Β H a - L D' N D N D0.2 0.4 0.6 0.8 1.0 Α H b - L D' N D N D 0.2 0.4 0.6 0.8 1.0 Β H b - L D' N D N D0.2 0.4 0.6 0.8 1.0 Α H c - L D' N D N D 0.2 0.4 0.6 0.8 1.0 Β H c - L D' N D N D FIG. 1. (color online). The measures D , D N and D ′ N as functions of α for the Werner state when (a-1) d = 2, (b-1) d = 3 and(c-1) d = 4, and that of the isotropic state when (a-2) d = 2, (b-2) d = 3 and (c-2) d = 4. For both the Werner state and theisotropic state, D N and D ′ N are monotonic functions of D . Werner states. The Werner states of a d ⊗ d dimensional system admit the form[50], ρ w = 2(1 − α ) d ( d + 1) Π + + 2 αd ( d − 1) Π − , α ∈ [0 , , (9)where Π + = ( I + F ) and Π − = ( I − F ) are projectors onto the symmetric and antisymmetric subspace of C d ⊗ C d respectively, F = P i,j | i a ih j a | ⊗ | j b ih i b | is the swap operator. Then D N ( ρ w ) = (1 − α ) , d = 2 , (1 − α ) , d = 3 , (3 − α ) , d = 4 , (10)and D ′ N ( ρ w ) = √ (1 − α ) , d = 2 , √ (1 − α ) , d = 3 , √ (3 − α ) , d = 4 . (11)The three measures of quantum correlation, i.e., D N , D ′ N and D , are illustrated in (a-1), (b-1) and (c-1) in Fig. 1 forcomparison, which reveals that the curves for D N and D ′ N have the same tendencies as that of D . Isotropic states. For the d ⊗ d isotropic state ρ is = 1 d − − β ) I + ( d β − P + ) , β ∈ [0 , , (12)where P + = d P i,j | i a ih j a | ⊗ | i b ih j b | is the maximally entangled pure state in C d ⊗ C d . Then D N ( ρ is ) = (1 − β ) , d = 2 , | − β | ( | − β | + | − β ) | ) , d = 3 , | − β | ( | − β | + | (1 − β | ) , d = 4 (13)and D ′ N ( ρ is ) = √ (1 − β ) , d = 2 , | − β | ( √ | − β | + | − β ) | ) , d = 3 , | − β | ( √ | − β | + | (1 − β | ) , d = 4 (14)The three measures of quantum correlation, i.e., D N , D ′ N and D , are illustrated in (a-2), (b-2) and (c-2) in Fig. 1 forcomparison. We see from this figure that the curves for D N and D ′ N have the same tendencies as that of D . It alsoimplies that i) for both the Werner states and the isotropic states, D N and D ′ N are close to each other, ii) D is closeto D N and D ′ N with increasing of the dimension d for the Werner states, which in contrast to that of the isotropicstates. Bell-diagonal states. The Bell-diagonal states for two-qubits can be written as σ ab = 14 ( I ⊗ I + X j =1 c j σ j ⊗ σ j ) = X a,b λ ab | β ab ih β ab | , (15)where the σ j s are Pauli operators, {| β ab i} are four Bell states | β ab i ≡ √ ( | , b i + ( − a | , ⊕ b i ). Then D N σ ab ) = 12 | c c | + | c | | c − c | + | c + c | ) , (16) D ′ N ( σ ab ) = 12 √ | c c | + | c |√ q c + c . (17)In Fig. 2, the level surfaces of D N and D ′ N are plotted respectively. By comparing them with that of D in Ref. [51],we find that the trends of D N and D ′ N are roughly the same as that of D : D N and D ′ N increase when D increasesroughly and vice versa. (The geometry of the set of the Bell-diagonal states is a tetrahedron with the four Bell statessit at the four vertices, the extreme points of tetrahedron (i.e., ( − , , , − , , , − 1) and ( − , − , − ρ = 12 | β ih β | + p | β ih β | + 1 − p | β ih β | , (18) ρ = p | β ih β | + 1 − p | β ih β | + | β ih β | ) , (19) ρ = p | β ih β | + (1 − p ) | β ih β | (20)and ρ = p | β ih β | + (1 − p ) | β ih β | . (21)The three measures of quantum correlation, i.e., D N , D ′ N and D , are compared in Fig. 3. For ρ , ρ and ρ , thevariation trends of D N and D ′ N coincide with that of D while for ρ the curves of D N and D ′ N have the same tendencyas that of D roughly. In addition, one can see that i) D N and D ′ N can both lager than and smaller than D , namely,there is no order relation between D and the two previous measures, ii) while the behavior of both measures D N and D ′ N is quite similar, they are quite different from that of D .Going further, we can quantify the symmetric quantum discord, i.e., the quantum discord up to both part A andpart B. Let {| k b i} be an orthonormal basis of H b , then any ρ acting on H a ⊗ H b admits the form ρ = X i,j E ij ⊗ B ij = X k,l A kl ⊗ F kl (22) FIG. 2. (color online). The surfaces of constant D N and D ′ N as functions of c , c and c for: (a) D N = 0 . 05, (b) D N = 0 . D N = 0 . 3; (a ′ ) D ′ N = 0 . 05, (b ′ ) D ′ N = 0 . ′ ) D ′ N = 0 . with F kl = | k b ih l b | . Here, A kl = Tr b ( a ⊗ | l b ih k b | ρ ) are local operators on H a . Let˜ D N ( ρ ) : = X i ≤ k,j ≤ l k [ B ij , B kl ] k + X i New measures of quantum discord has been proposed by means of the amount of the non-commutativity quantifiedby the trace norm and the Hilbert-Schmidt norm. Our method provides two calculable measures of quantum discordfrom a new perspective: unlike the original quantum discord and other quantum correlations were induced by somemeasurement, the two non-commutativity quantities we presented were not defined via measurements. Both of themcan be calculated directly for any state, avoiding the previous optimization procedure in calculation. The nullities ofour measures coincide with that of the original quantum discord and they are invariant under local unitary operationas well. The examples we analyzed indicate that, when comparing our quantities with the original quantum discord,although they are different and even have large difference for some special states, the non-commutativity measuresreflect the original quantity roughly overall. We can conclude, to a certain extent, that our approach can reflect theoriginal quantum discord for the set of states with arbitrary dimension. On the other hand, the non-commutativitymeasures reflect quantum discord in a larger scale than that of the original quantum discord, we thus can use thesemeasures to find quantum states with limited quantum discord or the maximal discordant states (especially for thestates represented by one or two parameters), etc.As usual, only the trace norm and the Hilbert-Schmidt norm are considered. In fact we can also use the generaloperator norm or other norms in the definitions of D N and D ′ N . 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