arXiv:0704.0969

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

Su Hu, Zong‐Wen Yu

Published Apr 7, 2007• 1 citation

Abstract

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 $\ketψ$, we prove $M$ is a compact operator. We also prove $\ketψ$ 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.

DOI: 10.48550/arxiv.0704.0969

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arXiv:0704.0969

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

Su Hu, Zong‐Wen Yu

Published Apr 7, 2007• 1 citation

arXiv HTML PDF

Abstract

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

\ketψ

, we prove

M

is a compact operator. We also prove

\ketψ

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.

DOI: 10.48550/arxiv.0704.0969

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Yu Jing Guo, Xiaofei Qi, Jinchuan Hou

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arXiv:quant-ph/0607195Quantum Information and ComputationSep 1, 200799 citations

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

\ketψ

, we prove

M

is a compact operator. We also prove

\ketψ

is separable if and only if

M

is a bounded linear operator with rank 1, that is the image of

M

In this paper, the realignment criterion and the RCCN criterion of separability for states in infinite-dimensional bipartite quantum systems are established. Let

H_A

and

H_B

be complex Hilbert spaces with

\dim H_A\otimes H_B=+\infty

. Let

ρ

be a state on

H_A\otimes H_B

and

\{δ_k\}

be the Schmidt coefficients of

ρ

as a vector in the Hilbert space

{\mathcal C}_2(H_A)\otimes{\mathcal C}_2(H_B)

. We introduce the realignment operation

ρ^R

and the computable cross norm

\|ρ\|_{\rm CCN}

ρ

and show that, if

ρ

is separable, then

\|ρ^{R}\|_{\rm Tr}=\|ρ\|_{\rm CCN}=\sum\limits_kδ_k\leq1.

In particular, if

ρ

is a pure state, then

ρ

is separable if and only if

\|ρ^{R}\|_{\rm Tr}=\|ρ\|_{\rm CCN}=\sum\limits_kδ_k=1