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

2010/09/01 by Guo, Yu, Hou, Jinchuan
#FOS: Mathematics #FOS: Physical sciences #Operator Algebras (math.OA) #Quantum Physics (quant-ph)

paper · doi:10.48550/arxiv.1009.0116

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⊗ H_B=+∞. Let ρ be a state on H_A⊗ H_B and \δ_k\ be the Schmidt coefficients of ρ as a vector in the Hilbert space \mathcal C₂(H_A)⊗\mathcal C₂(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}=∑_kδ_k≤1. In particular, if ρ is a pure state, then ρ is separable if and only if ‖ρ^R‖_{\rm Tr}=‖ρ‖_{\rm CCN}=∑_kδ_k=1.

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