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arXiv:0711.2865 · Physical Review A

Proper monogamy inequality for arbitrary pure quantum states

Yong-Cheng Ou, Heng Fan, Shao-Ming Fei

Published Jul 9, 2008• 82 citations
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Abstract

We provide a detailed proof of an analytical lower bound of entanglement quantified by concurrence for arbitrary bipartite quantum states. It is shown that though the bound does not allow one to detect bound entanglement, it is tight for some mixed states and can detect most of the free entanglement. On the other hand, it is known that the entanglement monogamy inequality proposed by Coffman, Kundu, and Wootters is, in general, not true for higher-dimensional quantum states. Inducing from the lower bound of concurrence, we find a proper form of entanglement monogamy inequality for arbitrary pure quantum states.

DOI
10.1103/physreva.78.012311

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We provide a detailed proof of an analytical lower bound of entanglement quantified by concurrence for arbitrary bipartite quantum states. It is shown that though the bound does not allow one to detect bound entanglement, it is tight for some mixed states and can detect most of the free entanglement. On the other hand, it is known that the entanglement monogamy inequality proposed by Coffman, Kundu, and Wootters is, in general, not true for higher-dimensional quantum states. Inducing from the lower bound of concurrence, we find a proper form of entanglement monogamy inequality for arbitrary pure quantum states.
We propose a method to detect genuine quantum correlation for multi-qubit pure states. We then derive a complementarity relations for pure quantum states of N qubits. We prove that in all many-qubit systems there exist strict monogamy laws for quantum correlations. On the other hand, it is known that the entanglement monogamy equality proposed by Coffman, Kundu, and Wootters is in general not true for multiqubit states. Inducing from the CKW equality, we find a proper form of entanglement monogamy equality for arbitrary quantum states. The total quantum correlation of qubit k with the remaining qubits Rk can be characterizes. Furthermore, the quantum correlation of qubit mn with the remaining qubits Rmn can also be obtained. Furthermore, some monogamy relations have been obtained.
Abstract In the present study, a trade off of sharing of entanglement between subsystems of a higher dimensional quantum state is derived. It is presented in terms of an inequality which is analogous to the Coffman–Kundu–Wootters inequality that succinctly describes monogamy of entanglement in dimensional pure state. To derive the monogamy inequality in dimension, G‐concurrence measure of entanglement is considered as a measure of entanglement of maximal dimension. The approach of the present study incidentally points toward a rigorous framework which enables to obtain an upper bound of G‐concurrence of a bipartite qudit mixed state. Obtained upper bound of G‐concurrence is then shown to satisfy a monogamy relation.
The monogamy of entanglement is generally discussed using a bipartite entanglement measure as an upper bound. Here we discuss a new kind of monogamous relation where the upper bound is given by a multipartite measure of entanglement, the generalized concurrence. We show a monogamous equality involving the multipartite concurrence, all the bipartite concurrences, and the genuine tripartite entanglement for pure three-qubit systems. The result extends to mixed states in an inequality involving the generalized concurrence and all the bipartite concurrences. We provide a counterexample showing that the result cannot be extended for systems with more than three qubits.
The monogamy inequality in terms of the concurrence, called the Coffman-Kundu-Wootters inequality [V. Coffman et al., Phys. Rev. A 61, 052306 (2000)], and its generalization [T. J. Osborne and F. Verstraete, Phys. Rev. Lett. 96, 220503 (2006)] hold on general nnn-qubit states including mixed ones. In this Brief Report, we consider the monogamy inequalities in terms of the fully entangled fraction and the teleportation fidelity. We show that the monogamy inequalities do not hold on general mixed states, while the inequalities hold on nnn-qubit pure states.