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Quantum Information Processing

Multipartite entanglement based on realignment moments

Hui Zhao, Shu-Ying Zhuang, Naihuan Jing, Mei-Ming Zhang

Published Mar 17, 2025• 1 citation
Publisher
DOI
10.1007/s11128-025-04713-w

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arXiv:1511.00375•Physical Review A•Oct 29, 2015•18 citations
Based on the realignment moments of density matrix, we study parameterized entanglement criteria for bipartite and multipartite states. By adjusting the different parameter values, our criterion can detect not only bound entangled states, but also non-positive partial transpose entangled states for bipartite quantum systems. Moreover, we propose the definition of multipartite realignment moments and generalize the result of bipartite systems to obtain a sufficient criterion to detect entanglement for multipartite quantum states in arbitrary dimensions. And we further improve the conclusion to obtain another new entanglement criterion. The new method can detect more entangled states than previous methods as backed by detailed examples.
Entanglement detection is an important problem in quantum information theory because quantum entanglement is a key resource in quantum information processing. Realignment criterion is a powerful tool for the detection of entangled states in bipartite and multipartite quantum systems. It works well not only for negative-partial-transpose entangled states (NPTESs) but also for positive-partial-transpose entangled states (PPTESs). Since the matrix corresponding to the realignment map is indefinite, the experimental implementation of the map is an obscure task. In this work, first, we approximate the realignment map to a positive map using the method of structural physical approximation, and then we show that the structural physical approximation of the realignment map (SPA-R) is completely positive. Positivity of the constructed map is characterized using moments which can be physically measured. Next, we develop a separability criterion based on our SPA-R map in the form of an inequality and show that the developed criterion not only detect NPTESs but also PPTESs. Further, we show that for a special class of states called Schmidt-symmetric states, the SPA-R separability criteria reduce to the original form of the realignment criteria. We provide some examples to support the results obtained. Moreover, we analyze the error that may occur because of approximating the realignment map.
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.