The Necessary and Sufficient Conditions of Separability for Bipartite Pure States in Infinite Dimensional Hilbert Spaces

Details The Necessary and Sufficient Conditions of Separability for Bipartite Pure States in Infinite Dimensional Hilbert Spaces The Necessary and Sufficient Conditions of Separability for Bipartite Pure States in Infinite Dimensional Hilbert Spaces

2007-04-07

arxiv.org/abs/0704.0969v1

Physics

Quantum Physics

4 pages

Scientific paper

In this paper, we present the necessary and sufficient conditions of separability for bipartite pure states in infinite dimensional Hilbert spaces. Let $M$ be the matrix of the amplitudes of $\ket\psi$, we prove $M$ is a compact operator. We also prove $\ket\psi$ is separable if and only if $M$ is a bounded linear operator with rank 1, that is the image of $M$ is a one dimensional Hilbert space. So we have related the separability for bipartite pure states in infinite dimensional Hilbert spaces to an important class of bounded linear operators in Functional analysis which has many interesting properties.

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