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Physical Review A

Analytically solvable two-qubit entanglement monotone

Jing Zhang, Chunwen Li, Tzyh‐Jong Tarn, Jianwu Wu

Published Sep 7, 2007• 2 citations
Publisher

Abstract

A so-called quasientanglement measure was obtained [J. Zhang et al., Phys. Rev. A 73, 022319 (2006)] by extending Jaeger's Minkowskian norm entanglement measure to general multipartite mixed states. The obtained quasientanglement measure has good properties, but it is still left open whether it is nonincreasing under general local operations and classical communications (LOCC). We show that the obtained quasientanglement measure is indeed nonincreasing on average under LOCC at least for two-qubit states. Thus, it is a good candidate for the two-qubit entanglement monotone. Further, it is shown that this entanglement measure is less than the concurrence for a large class of mixed entangled states.

DOI
10.1103/physreva.76.032306

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A so-called quasientanglement measure was obtained [J. Zhang et al., Phys. Rev. A 73, 022319 (2006)] by extending Jaeger's Minkowskian norm entanglement measure to general multipartite mixed states. The obtained quasientanglement measure has good properties, but it is still left open whether it is nonincreasing under general local operations and classical communications (LOCC). We show that the obtained quasientanglement measure is indeed nonincreasing on average under LOCC at least for two-qubit states. Thus, it is a good candidate for the two-qubit entanglement monotone. Further, it is shown that this entanglement measure is less than the concurrence for a large class of mixed entangled states.
In this Brief Report, we prove that an analytically solvable two-qubit entanglement monotone introduced by Zhang et al. [Phys. Rev. A 76, 032306 (2007)] is no greater than the square of concurrence. This proof is based on the equivalence between Zhang et al.'s monotone and an observable entanglement measure for mixed states suggested by Mintert and Buchleitner [Phys. Rev. Lett. 98, 140505 (2007)]. Our result may clarify the connections between several recently defined entanglement measures and help understand their possible applications in quantum-information theory.
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We develop an entanglement measure by extending Jaeger's Minkowskian norm entanglement measure. This measure can be applied to a much wider class of multipartite mixed states, although still ``quasi'' in the sense that it is still incapable of dividing precisely the sets of all separable and entangled states. As a quadratic scalar function of the system density matrix, the quasimeasure can be easily expressed in terms of the so-called coherence vector of the system density matrix, by which we show the basic properties of the quasimeasure including (1) zero entanglement for all separable states, (2) invariance under local unitary operations, and (3) nonincreasing under local positive operator-valued measure measurements. These results open up perspectives in further studies of dynamical problems in open systems, especially the dynamic evolution of entanglement, and the entanglement preservation against the environment-induced decoherence effects.
We present a measure of quantum entanglement which is capable of quantifying the degree of entanglement of a multi-partite quantum system. This measure, which is based on a generalization of the Schmidt rank of a pure state, is defined on the full state space and is shown to be an entanglement monotone, that is, it cannot increase under local quantum operations with classical communication and under mixing. For a large class of mixed states this measure of entanglement can be calculated exactly, and it provides a detailed classification of mixed states. We also present a connection to the best separable approximation of an entangled state of a two-qubit system.
We present a measure of quantum entanglement which is capable of quantifying the degree of entanglement of a multipartite quantum system. This measure, which is based on a generalization of the Schmidt rank of a pure state, is defined on the full state space, and is shown to be an entanglement monotone, that is, it cannot increase under local quantum operations with classical communication and under mixing. For a large class of mixed states this measure of entanglement can be calculated exactly, and it provides a detailed classification of mixed states.