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

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 $\|\rho^{R}\|_{\rm Tr}=\|\rho\|_{\rm CCN}=\sum\limits_k\delta_k=1$.