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Comparison of different measures for quantum discord under non-Markovian noise

Two geometric measures for quantum discord were recently proposed by Modi et al. [Phys. Rev. Lett. 104, 080501 (2010)] and Dakic et al. [Phys. Rev. Lett. 105, 190502 (2010)]. We study the similarities and differences for total quantum correlations of Bell-diagonal states using these two geometry-based quantum discord and the original quantum discord. We show that, under non-Markovian dephasing channels, quantum discord and one of the geometric measures stay constant for a finite amount of time, but not the other geometric measure. However, all the three measures share a common sudden change point. Our study on critical point of sudden transition might be useful for keeping long time total quantum correlations under decoherence.

Entangled states cannot be prepared by local operations and classical communication [1,2]. One may think that the exchange of classical information would not add any quantum correlation to the state. This is true for pure states but not for a general mixed state, because quantum correlations also exist in some mixed separable states and have played important roles in some quantum tasks, such as in deterministic quantum computation with one pure qubit [3]. To capture the total quantum correlations, a measure called quantum discord has been first proposed by Olliver and Zurek [4] and by Henderson and Vedral [5], and then widely studied .
Quantum discord is spoiled due to unavoidable interaction between the quantum system and the surrounding environment. The dynamics of quantum discord has been investigated under both Markovian [18][19][20][21][22][23] and non-Markovian [24] environments. Of particular interest is that, for a special class of quantum Bell-diagonal states, there exists sudden change of quantum discord under Markovian environment [19]. Moreover, the constant quantum discord under Markovian phase-damping channels was observed experimentally [20] and intensively studied theoretically [21,22].
Quantum discord (capturing total quantum correlations) has also been explored from the aspect of geometry, where two measures were recently proposed based on, respectively, the relative entropy [25] and the square of Hilbert-Schmidt norm [26]. The present work concentrates on the comparison of these two geometric measures with the originally defined quantum discord. Since some interesting features, such as suddenly changing and constant total quantum correlations, have been discovered by the quantum discord with respect to Bell-diagonal states, we wonder if these features remain in the two geometric measures. We will focus on non-Markovian environments, from which we could obtain some insights into the protection of total quantum correlations from decoherence.
Quantum discord is defined as a measure of the discrepancy between two different quantum analogs of the classical mutual information [4,5]. For a bipartite system ρ AB , the quantum discord is given by where I (ρ AB ) = S (ρ A )+S (ρ B )−S (ρ AB ) is the quantum mutual information (also called total correlations [5]), with S (ρ) = −Tr(ρ log 2 ρ) the von Neumann entropy of ρ and ρ A(B) the reduced density matrix of ρ AB by tracing out B(A).
is considered as the classical correlation, where ρ k A =Tr B B k ρ AB B † k /q k is the resulting state after the measurement {B k } on B, and q k =Tr AB B k ρ AB B † k . Note that quantum discord is not symmetric with respect to exchanging A and B, however, for the Bell-diagonal states under consideration, it is [26].
From the relative entropy perspective, the geometric measure Q R quantifies how distinguishable a given state ρ is from the closest classical state υ [25], where G is the set of classical states [28] and S (ρ||υ) = −Tr(ρ log 2 υ) − S (ρ) is the relative entropy.
Another geometric measure Q S is defined based on the fact that almost all quantum states have non-vanishing quantum discord [12,26]. The distance between a given state ρ and the nearest zero-discord state ̺ is defined as [26], where Ω denotes the set of quantum states with zero-discord [28] and ||ρ−̺|| 2 =Tr(ρ−̺) 2 is the square of Hilbert-Schmidt norm of Hermitian operators [26,27]. For a two-qubit system , with 1 and {σ j } being the identity and Pauli operators, Eq. (3) can be simplified as where α = (α 1 , α 2 , α 3 ) T is a column vector, M = (M jk ) is a matrix, and δ max is the largest eigenvalue of matrix α α T +MM T . Here the superscript T represents the transpose of vectors or matrices.
We start our analysis by considering two identical qubits A and B initially in a Belldiagonal state with the density operator as where the eigenstates are four Bell states |χ ab = (|0, b +(−1) a |1, 1⊕b )/ √ 2 with eigenvalues In what follows, we consider the situation of the qubits under independent non-Markovian dephasing channels [30]. The dynamics of the qubits can be characterized by the Kraus where the Kraus operators satisfy The Kraus operators for this non-Markovian model are given by tal noise bandwidth and Γ is the Markovian decay rate. Explicitly, the time evolution of the system can be expressed as ρ For simplicity, we denote in the following by c 2 (t) = ǫc 1 (t), with ǫ = c 2 (0)/c 1 (0). In addition, the above results could return to the Markovian situation by setting f (t) → Γt/2 in the Markovian limit γ → ∞.
In Fig. 1(c), we have plotted the critical point T (= Γτ ) as a function of dimensionless scaled reservoir noise bandwidth γ/Γ. We found that the critical point T grows with the decrease of the reservoir bandwidth. This implies that the non-Markovian behavior would prolong the quantum correlation under decoherence. When γ → ∞ (Markovian limit), τ reduces to the cases studied in Ref. [21].
This phenomenon is quite different from the cases measured by D or Q R . Although they share a common critical point τ , the original 'decoherence-free' area of total quantum correlations discovered by D or Q R [20][21][22] does not appear in the measure of Q S .
To be more clarified, we have plotted in Fig. 2  and c 2 (0) = −0.4, the trajectory of the Bell-diagonal states under decoherence is depicted as the red lines in Fig. 2. Clearly, the red line coincides with the straight contour line in Fig.   2(a), which means the quantum discord will not be spoiled in this 'decoherence-free' area [20][21][22]. In addition, there are three other 'decoherence-free' areas (See the black straight contour lines, one for ǫ = −c 3 and the other two for ǫ = −1/c 3 [31]) as depicted in Fig. 2(a).
For other values of ǫ (initial conditions), the red line will always cross the contour lines and no constant quantum discord will take place. However, in Fig. 2(b), the red straight lines always definitely go through the contour lines, which is a direct illustration of no constant quantum discord by Q S .
Since all the three quantities are to measure the total quantum correlations, we wonder why Q S is incompatible with D and Q R in describing the dynamics of Bell-diagonal states under non-Markovian dephasing channels? As both Q R and Q S are defined from geometric perspective, we guess the imcompatibility is from the fact that the nearest zero-discord state, belonging to an arbitrary Bell-diagonal state and measured by the square of Hilbert-Schmidt norm, is different from the closest classical state quantified by the relative entropy.
Since for an arbitrary Bell-diagonal state, the nearest zero-discord state measured by Q S is just the closest classical state quantified by Q R , the discrepancy we discovered in this work must be resulted from the intrinsic nature of the square of Hilbert-Schmidt norm and the relative entropy.
To summarize, we have investigated quantum discord of Bell-diagonal states under decoherence by three different definitions. The differences and similarities by using the three measures have been presented and discussed. The study of critical point under non-Markovian environment might be helpful for prolonging total quantum correlations under decoherence.
Finally, it would be really interesting to further explore the 'decoherence-free' area of total quantum correlations, which appears in the originally defined quantum discord D and relative entropy based Q R , but disappears from the geometric perspective Q S . It may lead to a more fundamental quantum information problem, that is, which measure, D (Q R ) or Q S , is more accurate to characterize total quantum correlations?
This work is supported by the National Natural Science Foundation of China under Grant