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.

Quantum discord is initially introduced by Ollivier and Zurek 4 and by Henderson and Vedral 5 . 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 a ⊗ H b , the quantum discord of ρ (up to part B) is defined by is interpreted as the quantum mutual information, ρ ρ ρ = − S ( ): Tr( log ) is the von Neumann entropy, ρ ρ ρ 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 B 37 . Analytical expressions are known only for certain classes of states 15,16,20,[38][39][40][41][42][43][44][45] . Consequently, different versions (or measures) of quantum discord have been proposed 19,24,25,46,47 : the discord-like quantities in 46 , the geometric measure 47 , the Bures distance measure 24 and the trace norm geometric measure 19 , etc. Unfortunately, all of theses measures are difficult to compute since they also need the minimization or maximization scenario.
Let {|i a 〉 } be an orthonormal basis of H a . Then any state ρ acting on H a ⊗ H b can be represented by 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 ij (Note here that, the 'operation' Θ ij 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 B ij s are mutually commuting normal operators 47,48 . It follows that the non-commutativity of the local operators B ij s 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, B ij s. In the following, we propose our approach: the non-commutativity measures. We present

Results
The amount of non-commutativity. Let X and Y be arbitrarily given operators on some Hilbert space. . 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 ij }, N({B ij }).
Non-commutativity measure of quantum discord.
be a state acting on H a ⊗ H b 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., for any unitary operator U a/b acting on H a/b (this implies that D N and ′ D N are independent on the choice of the local orthonormal bases: for any ρ. By the definitions, it is clear that both D N and ′ D N can be easily calculated for any state. Let |ψ〉 be a pure state with Schmidt decomposition ψ λ whenever d = 2, + 2 2 whenever d = 3 and +  Fig. 3(a,b)). In addition, D N and ′ D N characterize quantum discord in a more large scale than that of D roughly. For the two-qubit pure state ψ λ = ∑ k k k k a b , we can also calculate that Werner states. The Werner states of a d ⊗ d dimensional system admit the form 50 ,

Isotropic states. For the d ⊗ d isotropic state
is the maximally entangled pure state in   ⊗ (1 4 ) , 2 , Bell-diagonal states. The Bell-diagonal states for two-qubits can be written as    Especially, we consider  The three measures of quantum correlation, i.e., D N , ′ D N and D, are compared in Fig. 3. For ρ 1 , ρ 3 and ρ 4 , the variation trends of D N and ′ D N coincide with that of D while for ρ 2 the curves of D N and ′ D N have the same tendency as 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 and part B. Let {|k b 〉 } be an orthonormal basis of H b , then any ρ acting on H a ⊗ H b admits the form

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 N and ′ D N . In addition, Fig. 2 shows that the level surfaces of ′ D N 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 directions 51 ). 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.