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arXiv:2603.13648 · Journal of Physics A Mathematical and Theoretical

Quantum correlations in mutually unbiased bases and non-Markovian dephasing

Hermann L. Albrecht, David M. Bellorin

quant-phSubmitted Mar 13, 2026• Announced Mar 17, 2026• 0 citations• Revised Aug 12, 2026 (v2)
v1 · Mar 13, 2026v2 · Aug 12, 2026
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Abstract

Wu et al. introduced a symmetric measure $Q^{s}$ in terms of mutually unbiased bases (MUBs) that quantifies residual quantum correlations (RQC). The definition of this measure involves determining two complementary local bases via an optimization of the quantum mutual information. One defines the first local bases, the optimal computational ones, by maximizing the classical correlations measure. Next, one defines a new set of local bases that are complementary to the first ones. In the latter, one also calculates the corresponding measure of quantum correlations by maximization. Considering alternative optimizations of the mentioned measures, one can define other quantifiers. In particular, Mundarain and Ladrón de Guevara introduced local available quantum correlations (LAQC, $\mathcal{L}$) in terms of the complementary local bases to the ones defining the least classically correlated state. That is, they characterize LAQC by defining the optimal bases as those that minimize the classical correlations measure. This type of quantum correlation is closely related to an entanglement activation protocol proposed by Piani et al. In previous articles, we derived an exact solution for the LAQC measure for 2-qubit X states. Based on these previous results and methodology, we determine the corresponding exact analytical expression for $Q^s$ of 2-qubit X states. We then analyze the behavior of these two measures for two non-Markovian quantum dephasing channels: Random Telegraph (RT) and Modified Ornstein-Uhlenbeck (MOU) noises. We derive general conditions for sudden death and revival of these measures in X states and illustrate these results with four families of bipartite qubit states: Werner states, Maximally Nonlocal Mixed States (MNMS), Maximally Entangled Mixed States (MEMS), and a mixed symmetric separable state.

Journal reference
J. Phys. A: Math. Theor. 59, 365302 (2026)
DOI
10.1088/1751-8121/ae9dc1
Comments
14 pages, 10 figures, 2 Appendix

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Wu et al. introduced a symmetric measure QsQ^{s}Qs in terms of mutually unbiased bases (MUBs) that quantifies residual quantum correlations (RQC). The definition of this measure involves determining two complementary local bases via an optimization of the quantum mutual information. One defines the first local bases, the optimal computational ones, by maximizing the classical correlations measure. Next, one defines a new set of local bases that are complementary to the first ones. In the latter, one also calculates the corresponding measure of quantum correlations by maximization. Considering alternative optimizations of the mentioned measures, one can define other quantifiers. In particular, Mundarain and Ladrón de Guevara introduced local available quantum correlations (LAQC, L\mathcal{L}L) in terms of the complementary local bases to the ones defining the least classically correlated state. That is, they characterize LAQC by defining the optimal bases as those that minimize the classical correlations measure. This type of quantum correlation is closely related to an entanglement activation protocol proposed by Piani et al. In previous articles, we derived an exact solution for the LAQC measure for 2-qubit X states. Based on these previous results and methodology, we determine the corresponding exact analytical expression for QsQ^sQs of 2-qubit X states. We then analyze the behavior of these two measures for two non-Markovian quantum dephasing channels: Random Telegraph (RT) and Modified Ornstein-Uhlenbeck (MOU) noises. We derive general conditions for sudden death and revival of these measures in X states and illustrate these results with four families of bipartite qubit states: Werner states, Maximally Nonlocal Mixed States (MNMS), Maximally Entangled Mixed States (MEMS), and a mixed symmetric separable state.
Local available quantum correlations (LAQC), as defined by Mundarain et al., are analyzed for two-qubit X states with local Bloch vectors of equal magnitude. Symmetric X-states are invariant under the exchange of subsystems, hence having the same Bloch vector. On the other hand, anti-symmetric X states have Bloch vectors with an equal magnitude but opposite direction. In both cases, we obtain exact analytical expressions for their LAQC quantifier. We present some examples and compare this quantum correlation to concurrence and quantum discord. We have also included Markovian decoherence, with Werner states under amplitude damping decoherence. As is the case for depolarization and phase damping, no sudden death behavior occurs for the LAQC of these states with this quantum channel.
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Local available quantum correlations (LAQCs), as dened by Mundarain et al. [19], are analytically determined for Bell Diagonal states. Using the Kraus operators formalism [10], we analyze the dissipative dynamics of 2-qubit LAQCs under Markovian decoherence. This is done for Werner states under the depolarizing [20] and phase damping channels [21]. Since Werner states are among those that exhibit the so called entanglement sudden death [27], the results are compared with the ones obtained for Quantum Discord [22], as analyzed by Werlang et al. [24], as well as for entanglement, i.e. Concurrence[7]. The LAQCs quantier, as Quantum Discord does, only vanishes asymptotically.
We introduce measures of multipartite quantum correlations based on classical correlations in mutually unbiased bases. These classical correlations are measured in terms of the classical mutual information, which has a clear operational meaning. We present sufficient conditions under which these measures are maximized. States that have previously been shown to be particularly important in the context of local operations and classical communication fulfill these conditions and therefore maximize the correlation measures. In addition, we show how the introduced measures can be used to detect high-dimensional tripartite entanglement by using only a few local measurements.
The complementary quantum correlations (CQC) relation bounds the sum of two classical mutual informations, obtained from local mutually unbiased measurements, by the quantum mutual information of the premeasurement state. We refute it. Separable rank-two counterexamples exist in every local dimension pair m×nm\times nm×n with m,n≥3m,n\ge3m,n≥3, with closed-form excess at least 1/(8m2n2)1/(8m^2n^2)1/(8m2n2) nats, and in every qubit--qudit pair 2×n2\times n2×n with n≥3n\ge3n≥3 except n=3,5n=3,5n=3,5; every covered pair also admits full-rank separable counterexamples. We then analyse the two residual qubit--qudit dimensions. A dimension-free entropy envelope replaces the natural quadratic majorant and lowers the requirement for closing the equal-prior orthogonal two-ray family from a triangular-discrimination bound S≤8/3S\le8/3S≤8/3 to S≤3.8265583…S\le3.8265583\ldotsS≤3.8265583…; the associated gate matrix has trace exactly two, so its spectral test collapses to a single eigenvalue-free scalar; and the exact identity X=1−4Var⁡(c)X=1-4\operatorname{Var}(c)X=1−4Var(c) turns the prime-Fourier full-spark barrier into a variance bound. A 128128128-bit interval cover then closes that family at 2×32\times32×3 with gap at least 0.0120210.0120210.012021 nats, and at 2×52\times52×5 an exact saturator attaining S=(14+25)/5S=(14+2\sqrt5)/5S=(14+25​)/5 refutes three competing routes. We also give a state-dependent corrected inequality that is universal, and show it is incomparable with CQC already at 2×22\times22×2. Finally we quantify why the original searches had essentially no power to find these states: the violating set is a sliver against the low-rank boundary, the witness lies 7.37.37.3 standard deviations below the Hilbert--Schmidt mean, the bases must be aligned to about nine degrees (∼10−21\sim10^{-21}∼10−21 of frames), and extrapolating the sample minimum demands 101010^{10}1010 to 101810^{18}1018 samples against the 10710^7107 ever run.
An essential feature of genuine quantum correlation is the simultaneous existence of correlation in complementary bases. We reveal this feature of quantum correlation by defining measures based on invariance under a basis change. For a bipartite quantum state, the classical correlation is the maximal correlation present in a certain optimum basis, while the quantum correlation is characterized as a series of residual correlations in the mutually unbiased bases. Compared with other approaches to quantify quantum correlation, our approach gives information-theoretical measures that directly reflect the essential feature of quantum correlation.