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 Hₐ and HB be complex Hilbert spaces with Hₐ⊗ HB=+∞. Let ρ be a state on Hₐ⊗ HB and {δₖ} be the Schmidt coefficients of ρ as a vector in the Hilbert space C₂(Hₐ)⊗C₂(HB). We introduce the realignment operation ρʳ and the computable cross norm ρ CCN of ρ and show that, if ρ is separable, then ρʳ Tr=ρ CCN=∑ₖδₖ≤1. In particular, if ρ is a pure state, then ρ is separable if and only if ρʳ Tr=ρ CCN=∑ₖδₖ=1.