Abstract
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.
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Introduction
The characterization of quantum correlations in composite quantum states is of great importance in quantum information theory1,2,3,4,5,6. 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 bases8,9 and quantum correlation derived from the distance between the reduced states10, etc. Among them, quantum discord has aroused great interest in the past decade11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30. It is more robust against the effects of decoherence13 and can be a resource in quantum computation31,32, quantum key distribution33 remote state preparation34,35 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 Ha ⊗ Hb, the quantum discord of ρ (up to part B) is defined by
where, the minimum is taken over all local von Neumann measurements Πb, is interpreted as the quantum mutual information, is the von Neumann entropy, , and with , k = 1, 2, …, dim Hb. 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 states15,16,20,38,39,40,41,42,43,44,45. Consequently, different versions (or measures) of quantum discord have been proposed19,24,25,46,47: 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 {|ia〉} be an orthonormal basis of Ha. Then any state ρ acting on Ha ⊗ Hb can be represented by
where Eij = |ia〉〈ja| 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 Bij (Note here that, the ‘operation’ Θij 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 Bijs are mutually commuting normal operators47,48. It follows that the non-commutativity of the local operators Bijs 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, Bijs. 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 = {Ai : 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 {Bij}, N({Bij}).
Non-commutativity measure of quantum discord
Let be a state acting on Ha ⊗ Hb 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 Θijs on her part, 1 ≤ i, j ≤ dim Ha, then Bob can calculate the amount of non-commutativity through the reduced operators Bijs. By definition, it is obvious that i) DN(ρ) ≥ 0, , both DN and vanish only for the zero quantum discord states, i.e., iff D(ρ) = 0; ii) both DN and are invariant under the local unitary operations as that of the quantum discord, i.e., and for any unitary operator Ua/b acting on Ha/b (this implies that DN and are independent on the choice of the local orthonormal bases: if with respect to the local orthonormal basis {|ia〉 |jb〉} and with respect to another local orthonormal basis , then and for some local unitary operators Ua and Ub); iii) for any ρ. By the definitions, it is clear that both DN 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, DN(|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. DN and reach the maximum values only on the maximally entangled one.
It is worth mentioning here that both DN 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 DN(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 DN(ρ) and ). The continuity of DN 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 DN and . That is, all the associated conditions that satisfied by the original quantum discord are met by our quantities. From this perspective, DN 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 DN 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 DN and reflect the amount of quantum discord roughly: DN and increase (resp. decrease) if and only if D increase (resp. decrease) for almost all these states (see Figs 1, 2, 3). DN ≥ D and for almost all these states while there do exist states such that DN < D and (see Fig. 3(a,b)). In addition, DN 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.
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., DN, and D, are illustrated in (a-1), (b-1) and (c-1) in Fig. 1 for comparison, which reveals that the curves for DN 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., DN, 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 DN and have the same tendencies as that of D. It also implies that i) for both the Werner states and the isotropic states, DN and are close to each other, ii) D is close to DN 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 σjs are Pauli operators, {|βab〉} are four Bell states . Then
In Fig. 2, the level surfaces of DN and are plotted respectively. By comparing them with that of D in ref. 51, we find that the trends of DN and are roughly the same as that of D: DN 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., DN, and D, are compared in Fig. 3. For ρ1, ρ3 and ρ4, the variation trends of DN and coincide with that of D while for ρ2 the curves of DN and have the same tendency as that of D roughly. In addition, one can see that i) DN 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 DN 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 {|kb〉} be an orthonormal basis of Hb, then any ρ acting on Ha ⊗ Hb admits the form
with Fkl = |kb〉〈lb|. Here, Akl = Trb(1a ⊗ |lb〉〈kb|ρ) are local operators on Ha. 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 pij ≥ 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 DN 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).
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Acknowledgements
This work is supported by the National Natural Science Foundation of China under Grant No. 11301312.
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Guo, Y. Non-commutativity measure of quantum discord. Sci Rep 6, 25241 (2016). https://doi.org/10.1038/srep25241
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