Non-commutativity measure of quantum discord.

Quantum discord is a manifestation of quantum correlations due to non-commutativity rather than entanglement. Two measures of quantum discord by the amount of non-commutativity via the trace norm and the Hilbert-Schmidt norm respectively are proposed in this paper. These two measures can be calculated easily for any state with arbitrary dimension. It is shown by several examples that these measures can reflect the amount of the original quantum discord.

The characterization of quantum correlations in composite quantum states is of great importance in quantum information theory123456. It has been shown that there are quantum correlations that may arise without entanglement, such as quantum discord (QD)4, measurement-induced nonlocality (MIN)6, quantum deficit7, quantum correlation induced by unbiased bases89 and quantum correlation derived from the distance between the reduced states10, etc. Among them, quantum discord has aroused great interest in the past decade1112131415161718192021222324252627282930. It is more robust against the effects of decoherence13 and can be a resource in quantum computation3132, quantum key distribution33 remote state preparation3435 and quantum cryptography36.

Quantum discord is initially introduced by Ollivier and Zurek4 and by Henderson and Vedral5. The idea is to measure the discrepancy between two natural yet different quantum analogs of the classical mutual information. For a state ρ of a bipartite system A + B described by Hilbert space H ⊗ H, the quantum discord of ρ (up to part B) is defined by

where, the minimum is taken over all local von Neumann measurements Π, is interpreted as the quantum mutual information, is the von Neumann entropy, , and with , k = 1, 2, …, dim H. Calculation of quantum discord given by Eq. (1) in general is NP-complete since it requires an optimization procedure over the set of all measurements on subsystem B37. Analytical expressions are known only for certain classes of states1516203839404142434445. Consequently, different versions (or measures) of quantum discord have been proposed1924254647: the discord-like quantities in46, the geometric measure47, the Bures distance measure24 and the trace norm geometric measure19, etc. Unfortunately, all of theses measures are difficult to compute since they also need the minimization or maximization scenario.

Let {|i〉} be an orthonormal basis of H. Then any state ρ acting on H ⊗ H can be represented by

where E = |i〉〈j| and . That is, assume that Alice and Bob share a state ρ, if Alice take an ‘operation’

on her part, then Bob obtains the local operator B (Note here that, the ‘operation’ Θ is not the usual quantum operation which admits the Kraus sum respresentation). Quantum discord is from non-commutativity: D(ρ) = 0 if and only if Bs are mutually commuting normal operators4748. It follows that the non-commutativity of the local operators Bs implies ρ contains quantum discord. The central aim of this article is to show that, for any given state written as in Eq. (2), its quantum discord can be measured by the amount of non-commutativity of the local operators, Bs. In the following, we propose our approach: the non-commutativity measures. We present two measures: the trace norm measure and the Hilbert-Schmidt norm one. Both of them can be calculated for any state directly via the Lie product of the local operators. We then analyze our quantities for the Werner state, the isotropic state and the Bell-diagonal state in which the original quantum discord have been calculated. By comparing our quantities 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 − YX = 0 if and only if ||[X, Y]|| = 0, ||·|| is any norm defined on the operator space. That is, ||[X, Y]|| ≠ 0 implies the non-commutativity of X and Y. In general, ||[X, Y]|| reflects the amount of the non-commutativity of X and Y. Furthermore, for a set of operators Γ = {A : 1 ≤ i ≤ n}, the total non-commutativity of Γ can be defined by

In ref. 49, N(Γ) is used for measure the ‘quantumness’ of a quantum ensemble Γ when ||·|| is the trace norm ||·||Tr, i.e., . We remark here that any norm can be used for quantifying the amount. It is a natural way that, for any state as in Eq. (2), the amount of its non-commutativity can be considered as the total non-commutativity of {B}, N({B}).

Non-commutativity measure of quantum discord

Let be a state acting on H ⊗ H as in Eq. (2). We define a measure of QD for ρ by

Similarly, we can define

where ||·||2 denotes the Hilbert-Schmidt norm, i.e., . That is, if Alice takes Θs on her part, 1 ≤ i, j ≤ dim H, then Bob can calculate the amount of non-commutativity through the reduced operators Bs. By definition, it is obvious that i) D(ρ) ≥ 0, , both D and vanish only for the zero quantum discord states, i.e., iff D(ρ) = 0; ii) both D and are invariant under the local unitary operations as that of the quantum discord, i.e., and for any unitary operator U acting on H (this implies that D and are independent on the choice of the local orthonormal bases: if with respect to the local orthonormal basis {|i〉 |j〉} and with respect to another local orthonormal basis , then and for some local unitary operators U and U); iii) for any ρ. By the definitions, it is clear that both D and can be easily calculated for any state.

Let |ψ〉 be a pure state with Schmidt decomposition . Then

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(|ψ〉〈ψ|) = 0 (or ) if and only if |ψ〉 is separable. For the maximally entangled state in a d ⊗ d system, it is straightforward that whenever d = 2, whenever d = 3 and 4 whenever d = 4, whenever d = 2, whenever d = 3 and whenever d = 4. D and reach the maximum values only on the maximally entangled one.

It is worth mentioning here that both D and are defined without measurement, so the way we used is far different from the original quantum discord and other quantum correlations (note that all the measures of the quantum correlations proposed now are defined by some distance between the state and the post state after some measurement). In addition, it is clear that D(ρ) and are continuous functions of ρ since both the trace norm and Hilbert-Schmidt norm are continuous. In28, 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 correlation proposed in28 (note that the condition (1-d) in28 is invalid for D(ρ) and ). The continuity of D and meets the reasonable property (2-a) (note: (2-b) and (2-c) are invalid since these two conditions are associated with measurement-induced correlation). (7) and (8) guarantee the debatable property (3-a). (3-c) and (3-d) are not satisfied as that of the original quantum discord while (3-b) is invalid for D and . That is, all the associated conditions that satisfied by the original quantum discord are met by our quantities. From this perspective, D and 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 and with quantum discord D for several classes of well-known states and plot the level surfaces for the Bell-diagonal states. These examples will show that D and reflect the amount of quantum discord roughly: D and increase (resp. decrease) if and only if D increase (resp. decrease) for almost all these states (see Figs 1, 2, 3). D ≥ D and for almost all these states while there do exist states such that D < D and (see Fig. 3(a,b)). In addition, D and characterize quantum discord in a more large scale than that of D roughly. For the two-qubit pure state , we can also calculate that whenever λ1 > a with a ≈ 0.3841 while whenever λ1 < a and whenever λ1 > b with b ≈ 0.4279 while whenever λ1 < b.

Figure 1

The measures D, D and as a function 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 the isotropic state, D and are monotonic functions of D.

Figure 2

The surfaces of constant D and as a function of c1, c2 and c3 for: (a) D = 0.05, (b) D = 0.1 and (c) D = 0.3; (a′) , (b′) and (c′) .

Figure 3

The measures D, D and as a function of p for (a) ρ1, (b) ρ2, (c) ρ3 and (d) ρ4.

Werner states

The Werner states of a d ⊗ d dimensional system admit the form50,

where and are projectors onto the symmetric and antisymmetric subspace of respectively, is the swap operator. Then

and

The three measures of quantum correlation, i.e., D, and D, are illustrated in (a-1), (b-1) and (c-1) in Fig. 1 for comparison, which reveals that the curves for D and have the same tendencies as that of D.

Isotropic states

For the d ⊗ d isotropic state

where is the maximally entangled pure state in . Then

and

The three measures of quantum correlation, i.e., D, and D, are illustrated in (a-2), (b-2) and (c-2) in Fig. 1 for comparison. We see from this figure that the curves for D and have the same tendencies as that of D. It also implies that i) for both the Werner states and the isotropic states, D and are close to each other, ii) D is close to D and with increasing of the dimension d for the Werner states, which in contrast to that of the isotropic states.

Bell-diagonal states

The Bell-diagonal states for two-qubits can be written as

where the σs are Pauli operators, {|β〉} are four Bell states . Then

In Fig. 2, the level surfaces of D and are plotted respectively. By comparing them with that of D in ref. 51, we find that the trends of D and are roughly the same as that of D: D and increase when D increases roughly and vice versa. (The geometry of the set of the Bell-diagonal states is a tetrahedron with the four Bell states sit at the four vertices, the extreme points of tetrahedron (i.e., (−1, 1, 1), (1, −1, 1), (1, 1, −1) and (−1, −1, −1)), see Fig. 1 in ref. 51 for detail.)

Especially, we consider

and

The three measures of quantum correlation, i.e., D, and D, are compared in Fig. 3. For ρ1, ρ3 and ρ4, the variation trends of D and coincide with that of D while for ρ2 the curves of D and have the same tendency as that of D roughly. In addition, one can see that i) D and 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 and 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 and part B. Let {|k〉} be an orthonormal basis of H, then any ρ acting on H ⊗ H admits the form

with F = |k〉〈l|. Here, A = Tr(1 ⊗ |l〉〈k|ρ) are local operators on H. Let

where ||·|| is the trace norm, or the Hilbert-Schmidt norm, or other norms. Then i) and if and only if it is a classical-classical state (ρ is called a classical-classical state if with p ≥ 0 and ); ii) is invariant under the local unitary operations. We can conclude that quantifies the amount of the symmetric quantum discord of ρ.

Discussion

New measures of quantum discord has been proposed by means of the amount of the non-commutativity quantified by the trace norm and the Hilbert-Schmidt norm. Our method provides two calculable measures of quantum discord from a new perspective: unlike the original quantum discord and other quantum correlations were induced by some measurement, the two non-commutativity quantities we presented were not defined via measurements. Both of them can be calculated directly for any state, avoiding the previous optimization procedure in calculation. The nullities of our measures coincide with that of the original quantum discord and they are invariant under local unitary operation as 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 measures reflect the original quantity roughly overall. We can conclude, to a certain extent, that our approach can reflect the original quantum discord for the set of states with arbitrary dimension. On the other hand, the non-commutativity measures reflect quantum discord in a larger scale than that of the original quantum discord, we thus can use these measures to find quantum states with limited quantum discord or the maximal discordant states (especially for the states 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 general operator norm or other norms in the definitions of D and . In addition, Fig. 2 shows that the level surfaces of are nearly symmetric up to the four Bell states directions, which is very close to that of the quantum discord D (the level surfaces of D are symmetric up to the four Bell states directions51). Also note that the Hilbert-Schmidt norm is more easily calculated than the trace norm one, we thus use the Hilbert-Schmidt norm measure in general.

Additional Information

How to cite this article: Guo, Y. Non-commutativity measure of quantum discord. Sci. Rep. 6, 25241; doi: 10.1038/srep25241 (2016).