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arXiv:quant-ph/0604123 · Journal of Physics A Mathematical and General

A simple spectral condition implying separability for states of bipartite quantum systems

M E Gabach Clément, G. A. Raggio

Published Jul 5, 2006• 2 citations
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Abstract

For two qubits and for general bipartite quantum systems, we give a simple spectral condition in terms of the ordered eigenvalues of the density matrix which guarantees that the corresponding state is separable.
DOI
10.1088/0305-4470/39/29/019

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A new hierarchy of separability conditions for bipartite states is obtained. All the conditions in the hierarchy are necessary for separability. The conditions are expressed in terms of higher powers of the density operator of the bipartite system and the reduced density operators of the subsystems. Violation of any of the conditions in the hierarchy implies entanglement. As a consequence of the hierarchy, another separability condition, expressed in terms of the eigenvalues of the density operators of the total system and subsystems, is obtained.
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For any unitarily invariant convex function F on the states of a composite quantum system which isolates the trace there is a critical constant C such that F(w)<= C for a state w implies that w is not entangled; and for any possible D > C there are entangled states v with F(v)=D. Upper- and lower bounds on C are given. The critical values of some F's for qubit/qubit and qubit/qutrit bipartite systems are computed. Simple conditions on the spectrum of a state guaranteeing separability are obtained. It is shown that the thermal equilbrium states specified by any Hamiltonian of an arbitrary compositum are separable if the temperature is high enough.