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arXiv:quant-ph/0205054

A note on "A Matrix Realignment Method for Recognizing Entanglement," quant-ph/0205017 v1

Oliver Rudolph

Published May 10, 2002• 3 citations
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Abstract

In quant-ph/0205017 v1 Chen, Wu and Yang formulated a necessary separability criterion based on a realignment method for matrices. This note is to point out that this criterion is identical to the necessary cross norm criterion previously put forward by this author.
DOI
10.48550/arxiv.quant-ph/0205054

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A necessary separability criterion that relates the structures of the total density matrix and its reductions is given. The method used is based on the realignment method [K. Chen and L. A. Wu, Quant. Inf. Comput. 3, 193 (2003)]. The separability criterion naturally generalizes the reduction separability criterion introduced independently in the previous work [M. Horodecki and P. Horodecki, Phys. Rev. A 59, 4206 (1999) and N. J. Cerf, C. Adami, and R. M. Gingrich, Phys. Rev. A 60, 898 (1999)]. In special cases, it recovers the previous reduction criterion and the recent generalized partial transposition criterion [K. Chen and L. A. Wu, Phys. Lett. A 306, 14 (2002)]. The criterion involves only simple matrix manipulations and can therefore be easily applied.
For a quantum state in a bipartite system represented as a density matrix, researchers used the realignment matrix and functions on its singular values to study the separability of the quantum state. We obtain bounds for elementary symmetric functions of singular values of realignment matrices. This answers some open problems proposed by Lupo, Aniello and Scardicchio. As a consequence, we show that the proposed scheme by these authors for testing separability would not work if the two subsystems of the bipartite system have the same dimension.
Separability problem, to decide whether a given state is entangled or not, is a fundamental problem in quantum information theory. We propose a powerful and computationally simple separability criterion, which allows us to detect the entanglement of many bound entangled states. The criterion is strictly stronger than the criterion based on Bloch representations, the computable cross-norm or realignment criterion and its optimal nonlinear entanglement witnesses. Furthermore, this criterion can be generalized to an analog of permutation separability criteria in the even-partite systems.