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arXiv:2204.07552 · International Journal of Modern Physics B

Local available quantum correlations of non-symmetric X-states

David Bellorín, Hermann L. Albrecht Q, Douglas F. Mundarain

Published Jul 19, 2022• 2 citations
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Abstract

Local available quantum correlations (LAQC), as defined by Mundarain et al., are analyzed for nonsymmetric 2-qubit X states, that is, X states that are not invariant under the exchange of subsystems and therefore have local Bloch vectors whose norms are different. A simple analytic expression for their LAQC quantifier is obtained. As an example, we analyze the local application of the amplitude damping channel for Werner states and general X states. Although this local quantumchannel can create quantum discord in some cases, no such outcome is possible for LAQC, which hints toward their monotonicity under LOCC operations. This work, along with our previous result for the so-called symmetric and anti-symmetric X states, completes the pursuit of exact analytical expressions for the LAQC quantifier for 2-qubit X states.

DOI
10.1142/s0217979222501545

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Local available quantum correlations (LAQC), as defined by Mundarain et al., are analyzed for nonsymmetric 2-qubit X states, that is, X states that are not invariant under the exchange of subsystems and therefore have local Bloch vectors whose norms are different. A simple analytic expression for their LAQC quantifier is obtained. As an example, we analyze the local application of the amplitude damping channel for Werner states and general X states. Although this local quantumchannel can create quantum discord in some cases, no such outcome is possible for LAQC, which hints toward their monotonicity under LOCC operations. This work, along with our previous result for the so-called symmetric and anti-symmetric X states, completes the pursuit of exact analytical expressions for the LAQC quantifier for 2-qubit X states.
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.
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 (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 a quantum Fisher information based measurement-induced nonlocality (QFI-MIN), which quantifies the maximal statistical distinguishability induced by locally invariant unitary dynamics. The proposed measure inherits desirable properties including positivity, local unitary invariance, monotonicity under local operations, and immunity to the local ancilla problem. Analytical expressions are obtained for pure states, arbitrary two-qubit states, and two-qubit X states, revealing a direct connection with entanglement for pure systems. We further establish clear operational interpretations of QFI-MIN in quantum parameter estimation, local channel discrimination, and correlation-assisted communication. Its behavior under amplitude damping, depolarizing, and generalized amplitude damping channels demonstrates robustness against environmental noise. The proposed framework provides a physically consistent and operationally meaningful quantifier of quantum correlations, linking nonlocality, quantum metrology, and quantum information processing
By using quantum discord (QD), measurement induced non-locality (MIN) and negativity (QE), quantum correlation and entanglement are investigated for two qubits in two different cases for the initial two qubit Werner states, taking into account the influence of qubit damping. It is shown that there is no asymptotic decay for MIN while asymptotic decay exists for QD and QE. Quantum correlations cannot be strengthened by introducing the damping. The appearance time of stationary correlations gets shorter with the increase in the damping parameter. Finally, a uniform damping qubit can affect the stationary correlations when the qubits are initially in an entangled state.
Quantum discord, a kind of quantum correlation, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. In general, this correlation is different from entanglement, and quantum discord may be nonzero even for certain separable states. Even in the simple case of bipartite quantum systems, this different kind of quantum correlation has interesting and significant applications in quantum information processing. So far, quantum discord has been calculated explicitly only for a rather limited set of two-qubit quantum states and expressions for more general quantum states are not known. In this article, we derive explicit expressions for quantum discord for a larger class of two-qubit states, namely, a seven-parameter family of so called XXX states that have been of interest in a variety of contexts in the field. We also study the relation between quantum discord, classical correlation, and entanglement for a number of two-qubit states to demonstrate that they are independent measures of correlation with no simple relative ordering between them.