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Journal of Mathematical Physics

Properties of Some Connected Kernels in Multiparticle Systems

J. M. Combes

Published Aug 1, 1971• 2 citations
Publisher

Abstract

Connectivity of products of many-particle operators is investigated. Connected products of compact and Hilbert—Schmidt operators are in the same class as their factors. However, even in the simplest case of rank-one many-particle factors, the connected products are not necessarily in the trace class, but simple sufficiency conditions are established.

DOI
10.1063/1.1665796

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Connectivity of products of many-particle operators is investigated. Connected products of compact and Hilbert—Schmidt operators are in the same class as their factors. However, even in the simplest case of rank-one many-particle factors, the connected products are not necessarily in the trace class, but simple sufficiency conditions are established.
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In this paper, we present the necessary and sufficient conditions of separability for bipartite pure states in infinite dimensional Hilbert spaces. Let MMM be the matrix of the amplitudes of ∣ψ⟩\ketψ∣ψ⟩, we prove MMM is a compact operator. We also prove ∣ψ⟩\ketψ∣ψ⟩ is separable if and only if MMM is a bounded linear operator with rank 1, that is the image of MMM 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.