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Reports on Mathematical Physics

Realignment Operation and CCNR Criterion of Separability for States in Infinite-Dimensional Quantum Systems

Yu Jing Guo, Jinchuan Hou

Published Aug 1, 2013• 10 citations
Publisher
DOI
10.1016/s0034-4877(14)60002-1

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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 HAH_AHA​ and HBH_BHB​ be complex Hilbert spaces with dim⁡HA⊗HB=+∞\dim H_A\otimes H_B=+\inftydimHA​⊗HB​=+∞. Let ρρρ be a state on HA⊗HBH_A\otimes H_BHA​⊗HB​ and {δk}\{δ_k\}{δk​} be the Schmidt coefficients of ρρρ as a vector in the Hilbert space C2(HA)⊗C2(HB){\mathcal C}_2(H_A)\otimes{\mathcal C}_2(H_B)C2​(HA​)⊗C2​(HB​). We introduce the realignment operation ρRρ^RρR and the computable cross norm ∥ρ∥CCN\|ρ\|_{\rm CCN}∥ρ∥CCN​ of ρρρ and show that, if ρρρ is separable, then ∥ρR∥Tr=∥ρ∥CCN=∑kδk≤1.\|ρ^{R}\|_{\rm Tr}=\|ρ\|_{\rm CCN}=\sum\limits_kδ_k\leq1.∥ρR∥Tr​=∥ρ∥CCN​=k∑​δk​≤1. In particular, if ρρρ is a pure state, then ρρρ is separable if and only if ∥ρR∥Tr=∥ρ∥CCN=∑kδk=1\|ρ^{R}\|_{\rm Tr}=\|ρ\|_{\rm CCN}=\sum\limits_kδ_k=1∥ρR∥Tr​=∥ρ∥CCN​=k∑​δk​=1.
By combining a parameterized Hermitian matrix, the realignment matrix of the bipartite density matrix, and the vectorization of its reduced density matrices, we present a family of separability criteria which are stronger than the computable cross-norm or realignment (CCNR) criterion. With linear contraction methods, the proposed criteria can be used to detect the multipartite entangled states that are biseparable under any bipartite partitions. Moreover, we show by examples that the presented multipartite separability criteria can be more efficient than the corresponding multipartite realignment criterion based on CCNR, multipartite correlation tensor criterion, and multipartite covariance matrix criterion.