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Hilbert–Schmidt Measure of Pairwise Quantum Discord for Three-Qubit X States

Mohammed Daoud A, R. Ahl Laamara, S. Seddik

Published Oct 1, 2015• 2 citations
OpenAlex

Abstract

The Hilbert-Schmidt distance between a mixed three-qubit state and its closest state is used to quantify the amount of pairwise quantum correlations in a tripartite system.Analytical expressions of geometric quantum discord are derived.A particular attention is devoted to two special classes of three-qubit X states.They include three-qubit states of W, GHZ and Bell type.We also discuss the monogamy property of geometric quantum discord in some mixed three-qubit systems.

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The Hilbert-Schmidt distance between a mixed three-qubit state and its closest state is used to quantify the amount of pairwise quantum correlations in a tripartite system.Analytical expressions of geometric quantum discord are derived.A particular attention is devoted to two special classes of three-qubit X states.They include three-qubit states of W, GHZ and Bell type.We also discuss the monogamy property of geometric quantum discord in some mixed three-qubit systems.
A recursive approach to determine the Hilbert–Schmidt measure of pairwise quantum discord in a special class of symmetric states of k qubits is presented. We especially focus on the reduced states of k qubits obtained from a balanced superposition of symmetric n-qubit states (multiqubit Schrödinger cat states) by tracing out n-k particles (k = 2, 3, …, n-1). Two pairing schemes are considered. In the first one, the geometric discord measuring the correlation between one qubit and the parity grouping (k-1) qubits is explicitly derived. This uses recursive relations between the Fano–Bloch correlation matrices associated with subsystems comprising k, k-1, … and two particles. A detailed analysis is given for two-, three- and four-qubit systems. In the second scheme, the subsystem comprising the (k-1) qubits is mapped into a system of two logical qubits. We show that these two bipartition schemes are equivalents in evaluating the pairwise correlation in multiqubits systems. The explicit expressions of classical states presenting zero discord are derived.
In this paper, we obtain the pairwise quantum discord for a three-qubit W-class state, and investigate the monogamy property of quantum discord by two different ways (relative entropy-based distance and geometric square-norm distance). Through numerical calculations, we find that a party cannot have maximal quantum correlations with the other two parties simultaneously. For the given state, the quantum correlation between particles 1 and 3 induces limitation on the quantum correlation between them and particle 2. Moreover, the result also shows that the geometric quantum discord of the given W-class state obeys the monogamy property while the entropy quantum discord violates.
Quantum correlations may be measured by means of the distance from the state to the subclass of states $\ensuremath{\Omega}$ having well-defined classical properties. In particular, a geometric measure of asymmetric discord [Daki\ifmmode \acute{c}\else \'{c}\fi{} et al., Phys. Rev. Lett. 105, 190502 (2010)] was recently defined as the Hilbert-Schmidt distance from a given two-qubit state to the closest classical-quantum (CQ) correlated state. We analyze a geometric measure of symmetric discord defined as the Hilbert-Schmidt distance from a given state to the closest classical-classical (CC) correlated state. The optimal member of $\ensuremath{\Omega}$ is just a specially measured original state for both the CQ and CC discords. This implies that this measure is equal to the quantum deficit of postmeasurement purity. We discuss some general relations between the CC discords and explain why an analytical formula for the CC discord, contrary to the CQ discord, can hardly be found even for a general two-qubit state. Instead of such an exact formula, we find simple analytical-measurement-based upper bounds for the CC discord which, as we show, are tight and faithful in the case of two qubits and may serve as independent indicators of two-party quantum correlations. In particular, we propose an adaptive upper bound, which corresponds to the optimal states induced by single-party measurements: optimal measurement on one of the parties determines an optimal measurement on the other party. We discuss how to refine the adaptive upper bound by nonoptimal single-party measurements and by an iterative procedure which usually rapidly converges to the CC discord. We also raise the question of optimality of the symmetric measurements realizing the CC discord on symmetric states and give a partial answer for the qubit case.
A symmetric measure of quantum correlation based on the Hilbert-Schmidt distance is presented in this paper. For two-qubit states, we simplify considerably the optimization procedure so that numerical evaluation can be performed efficiently. Analytical expressions for the quantum correlation are attained for some special states. We further investigate the dynamics of quantum correlation of the system qubits in the presence of independent dissipative environments. Several nontrivial aspects are demonstrated. We find that the quantum correlation can increase even if the system state is suffering dissipative noise. Sudden changes occur, even twice, in the time evolution of quantum correlation. There is certain correspondence between the evolution of quantum correlation in the systems and that in the environments, and the quantum correlation in the systems will be transferred into the environments completely and asymptotically.
A symmetric measure of quantum correlation based on the Hilbert—Schmidt distance is presented in this paper. For two-qubit states, we considerably simplify the optimization procedure so that numerical evaluation can be performed efficiently. Analytical expressions for the quantum correlation are attained for some special states. We further investigate the dynamics of quantum correlation of the system qubits in the presence of independent dissipative environments. Several nontrivial aspects are demonstrated. We find that the quantum correlation can increase even if the system state is suffering from dissipative noise. Sudden changes occur, even twice, in the time evolution of quantum correlation. There exists a certain correspondence between the evolution of quantum correlation in the systems and that in the environments, and the quantum correlation in the systems will be transferred into the environments completely and asymptotically.