Chinese Science Bulletin

Sufficient and necessary conditions of separability for bipartite pure states in infinite-dimensional systems

Yu Jing Guo, Xiaofei Qi, Jinchuan Hou

Published Mar 1, 2011• 21 citations

Abstract

The PHC criterion and the realignment criterion for pure states in infinite-dimensional bipartite quantum systems are given. Furthermore, several equivalent conditions for pure states to be separable are generalized to infinite-dimensional systems.

DOI: 10.1007/s11434-010-4500-x

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Chinese Science Bulletin

Sufficient and necessary conditions of separability for bipartite pure states in infinite-dimensional systems

Yu Jing Guo, Xiaofei Qi, Jinchuan Hou

Published Mar 1, 2011• 21 citations

Publisher

Abstract

DOI: 10.1007/s11434-010-4500-x

Related papers

Sufficient and necessary conditions of separability for bipartite pure states in infinite-dimensional systems

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Chinese Science BulletinJan 1, 20110 citations

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Su Hu, Zong‐Wen Yu

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.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.

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The RCCN criterion of separability for states in infinite-dimensional quantum systems

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

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Sufficient and necessary conditions of separability for bipartite pure states in infinite-dimensional systems

Guo Guo, Yu, Qi Qi +3 more

中国科学通报：英文版Jan 1, 20110 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

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.

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