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

Abstract: 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 $\rho$ be a state on $H_A\otimes H_B$ and ${\delta_k}$ be the Schmidt coefficients of $\rho$ as a vector in the Hilbert space ${\mathcal C}2(H_A)\otimes{\mathcal C}_2(H_B)$. We introduce the realignment operation $\rho^R$ and the computable cross norm $|\rho|{\rm CCN}$ of $\rho$ and show that, if $\rho$ is separable, then $|\rho^{R}|_{\rm Tr}=|\rho|{\rm CCN}=\sum\limits_k\delta_k\leq1.$ In particular, if $\rho$ is a pure state, then $\rho$ is separable if and only if $|\rho^{R}|{\rm Tr}=|\rho|_{\rm CCN}=\sum\limits_k\delta_k=1$.