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 $\Vert \rho {\Vert }_{\mathrm{CCN}}$ of $\rho$ and show that, if $\rho$ is separable, then $\Vert {\rho }^R{\Vert }_{\mathrm{Tr}}=\Vert \rho {\Vert }_{\mathrm{CCN}}=\sum _k{\delta }_k\le 1.$ In particular, if $\rho$ is a pure state, then $\rho$ is separable if and only if $\Vert {\rho }^R{\Vert }_{\mathrm{Tr}}=\Vert \rho {\Vert }_{\mathrm{CCN}}=\sum _k{\delta }_k=1$.

18 pages

References in corpus (4)

• Entanglement detection beyond the cross-norm or realignment criterion
• Constructing entanglement witnesses for infinite-dimensional systems
• Positive finite rank elementary operators and characterizing entanglement of states
• The Quantum Separability Problem for Gaussian States