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Advanced Quantum Technologies

Separability Criteria via Type‐II Discrete Cosine Transform

A. A. Xu, Baozhi Sun, Zhen Wang, Shao-Ming Fei

Published Sep 1, 2026• 0 citations
Publisher

Abstract

ABSTRACT Quantum entanglement is a fundamental resource in quantum information science. We present three effective separability criteria for arbitrary bipartite quantum states based on the Type‐II discrete cosine transform. The proposed framework unifies and significantly generalizes the computable cross‐norm and realignment criterion (CNNR), the de Vicente criterion, as well as a wide range of existing separability criteria formulated within the Bloch representation framework. Moreover, our approach admits a natural extension to the multipartite setting via the multipartite correlation matrix. Detailed examples demonstrate that our criteria outperform the existing ones in entanglement detection.

DOI
10.1002/qute.70453

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ABSTRACT Quantum entanglement is a fundamental resource in quantum information science. We present three effective separability criteria for arbitrary bipartite quantum states based on the Type‐II discrete cosine transform. The proposed framework unifies and significantly generalizes the computable cross‐norm and realignment criterion (CNNR), the de Vicente criterion, as well as a wide range of existing separability criteria formulated within the Bloch representation framework. Moreover, our approach admits a natural extension to the multipartite setting via the multipartite correlation matrix. Detailed examples demonstrate that our criteria outperform the existing ones in entanglement detection.
Separability problem, to decide whether a given state is entangled or not, is a fundamental problem in quantum information theory. We propose a powerful and computationally simple separability criterion, which allows us to detect the entanglement of many bound entangled states. The criterion is strictly stronger than the criterion based on Bloch representations, the computable cross-norm or realignment criterion and its optimal nonlinear entanglement witnesses. Furthermore, this criterion can be generalized to an analog of permutation separability criteria in the even-partite systems.
A family of separability criteria based on correlation matrix (tensor) is provided. Interestingly, it unifies several criteria known before like, e.g., computable cross-norm or realignment criterion (CNNR), de Vicente criterion, and derived recently separability criterion based on symmetric informationally complete positive operator valued measures (SIC POVMs). It should be stressed that, unlike the well-known correlation matrix criterion or criterion based on local uncertainty relations, our criteria are linear in the density operator and hence one may find unexplored classes of entanglement witnesses and positive maps. Interestingly, there is a natural generalization to multipartite scenario using multipartite correlation matrix. We illustrate the detection power of the above criteria on several well-known examples of quantum states.
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.
The detection and quantification of quantum entanglement are central problems in quantum information science. Based on the realignment of density matrices and the vectorization of reduced density matrices, we construct a family of enhanced separability criteria for bipartite and multipartite quantum systems. Our criteria recover the previously proposed realignment-based criteria as special cases and detect more entanglement than the earlier ones. Moreover, we derive tighter lower bounds for the concurrence and the convex-roof extended negativity, criteria for detecting the genuine tripartite entanglement, and lower bounds on genuine-multipartite-entanglement concurrence. The advantages of our results are demonstrated with detailed examples.
Abstract Based on the generalized Bloch representation, we study the separability and entanglement of arbitrary dimensional multipartite quantum states. Some sufficient and some necessary criteria are presented. For certain states, these criteria together are both sufficient and necessary. Detailed examples show that our criteria are better than some existing ones in identifying separability. Based on these criteria, the largest separable ball around the maximally mixed state for arbitrary multi-qubit systems is found, and it is proved that its radius defined via the l 1-norm of the Bloch vector is the constant 1. Furthermore, the criteria in this paper can be implemented experimentally.