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Journal of Physics B Atomic Molecular and Optical Physics

Geometric measure of quantum discord for superpositions of Dicke states

Xiaolei Yin, Zhengjun Xi, Xiao-Ming Lu, Zhe Sun, Xiaoguang Wang

Published Nov 29, 2011• 20 citations
Publisher

Abstract

The pairwise quantum correlations of symmetric multi-qubit states are studied by using the geometric measure of quantum discord (GMQD). The explicit expressions of the GMQD for the Dicke states and their superpositions are given. By comparing the GMQD with quantum discord, we conclude that they display similar behaviour. From the results, we know that most of the states we have considered exhibit pairwise quantum correlations.

DOI
10.1088/0953-4075/44/24/245502

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The pairwise quantum correlations of symmetric multi-qubit states are studied by using the geometric measure of quantum discord (GMQD). The explicit expressions of the GMQD for the Dicke states and their superpositions are given. By comparing the GMQD with quantum discord, we conclude that they display similar behaviour. From the results, we know that most of the states we have considered exhibit pairwise quantum correlations.
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Quantum discord(QD) can be used to characterize quantum correlation of a quantum system. Based on geometric measurement of QD(GMQD),quantum correlation of the single-parameter X- state was calculated,and the results for the case were also given under three different quantum channels.For the amplitude-damping channel,by comparing concurrence and GMQD of the X-state mentioned above using graphical method,it's concluded that GMQD exhibits more robust than concurrence when the quantum system is exposed to environment.
The dynamics of a geometric measure of the quantum discord (GMQD) under decoherence is investigated. We show that the GMQD of a two-qubit state can be alternatively obtained through the singular values of a 3×43\times43×4 matrix whose elements are the expectation values of Pauli matrices of the two qubits. By using Heisenberg picture, the analytic results of the GMQD is obtained for three typical kinds of the quantum decoherence channels. We compare the dynamics of the GMQD with that of the quantum discord and of entanglement. We show that a sudden change in the decay rate of the GMQD does not always imply that of the quantum discord, and vice versa. We also give a general analysis on the sudden change in behavior and find that at least for the Bell diagonal states, the sudden changes in decay rates of the GMQD and that of the quantum discord occur simultaneously.
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.