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Journal of Xianyang Normal University

A Note on the Separability of Quantum Mixed States

Cao Huai-xin

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

In this note,we give a representation of density matrix for a bipartite quantum system and a neces-sary condition for separability of density matrix of mixed states in a bipartite quantum system of arbitrary dimen-tion.We show that if density matrix of mixed states in a bipartite quantum system of arbitrary dimension is separa-ble,then the trace norm of constructive matrix is less than or equal to 1.

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In this note,we give a representation of density matrix for a bipartite quantum system and a neces-sary condition for separability of density matrix of mixed states in a bipartite quantum system of arbitrary dimen-tion.We show that if density matrix of mixed states in a bipartite quantum system of arbitrary dimension is separa-ble,then the trace norm of constructive matrix is less than or equal to 1.
Given a bipartite quantum system represented by a Hilbert space ${\mathcal{H}}_{1}\ensuremath{\bigotimes}{\mathcal{H}}_{2},$ we give an elementary argument to show that if either $\mathrm{dim}{\mathcal{H}}_{1}=\ensuremath{\infty}$ or $\mathrm{dim}{\mathcal{H}}_{2}=\ensuremath{\infty},$ then the set of nonseparable density operators on ${\mathcal{H}}_{1}\ensuremath{\bigotimes}{\mathcal{H}}_{2}$ is trace-norm dense in the set of all density operators (and the separable density operators nowhere dense). This result complements recent detailed investigations of separability, which show that when $\mathrm{dim}{\mathcal{H}}_{i}<\ensuremath{\infty}$ for i=1,2,i=1,2,i=1,2, there is a separable neighborhood (perhaps very small for large dimensions) of the maximally mixed state.
Generally,to decide a given quantum state to be entangled or not,or to be useful for quantum information processing purposes is a difficult question.In this note,the separability of quantum states in bipartite quantum systems is investigated.According to the analysis of the expanded form of density matrix in the separable quantum state and the relationship of expansion coefficient,a new separability criterion of bipartite quantum states is obtained: if a quantum state in a bipartite quantum system of arbitrary dimension is separable,the expansion coefficient satisfies certain inequality.This method requires only the dimensions of systems and the expansion coefficient of density matrix which can be easily got,and the computation is easy,so it has its application value.