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Violation of monogamy inequality for qutrits and higher dimensional objects

Yong Cheng Ou

Published Dec 15, 2006• 0 citations
OpenAlex

Abstract

Bipartite quantum entanglement for qutrits or higher dimensional objects is considered. In a not strict sense but with an explicit two-qutrit antisymmetric state example, it is proved that a monogamy inequality for qubits introduced by Coffman, Kundu, and Wootters may be violated for their entanglement quantified by the concurrence in qutrits or higher dimensional objects.

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•May 15, 2014•0 citations
Bipartite quantum entanglement for qutrits or higher dimensional objects is considered. In a not strict sense but with an explicit two-qutrit antisymmetric state example, it is proved that a monogamy inequality for qubits introduced by Coffman, Kundu, and Wootters may be violated for their entanglement quantified by the concurrence in qutrits or higher dimensional objects.
Bipartite quantum entanglement for qutrits and higher-dimensional objects is considered. We analyze the possibility of violation of monogamy inequality, introduced by Coffman, Kundu, and Wootters, for some systems composed of such objects. An explicit counterexample with a three-qutrit totally antisymmetric state is presented. Since three-tangle has been confirmed to be a natural measure of entanglement for qubit systems, our result shows that the three-tangle is no longer a legitimate measure of entanglement for states with three qutrits or higher-dimensional objects.
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.
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.
We investigate possible generalizations of the Coffman-Kundu-Wootters monogamy inequality to four qubits, accounting for multipartite entanglement in addition to the bipartite terms. We show that the most natural extension of the inequality does not hold in general, and we describe the violations of this inequality in detail. We investigate alternative ways to extend the monogamy inequality to express a constraint on entanglement sharing valid for all four-qubit states, and perform an extensive numerical analysis of randomly generated four-qubit states to explore the properties of such extensions.