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Laser Physics Letters

The enhanced separability criteria based on equiangular tight frames

Xianfei Qi, Haiyan Zhang

Published Dec 10, 2024• 1 citation
Publisher

Abstract

Abstract The detection of quantum entanglement is an essential issue in the theory of quantum information. Recently, an elegant separability criterion to detect the entanglement of arbitrary-dimensional bipartite states is presented in Shi (2024 J. Phys. A: Math. Theor. 57 075302) by applying the positive operator valued measurements based on the equiangular tight frames (ETFs). Here we derive two enhanced separability criteria for detecting bipartite entanglement in arbitrary-dimensional quantum states using ETFs. Furthermore, we prove that they are not weaker than the criterion proposed in Shi (2024 J. Phys. A: Math. Theor. 57 075302).

DOI
10.1088/1612-202x/ad97b0

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Abstract The detection of quantum entanglement is an essential issue in the theory of quantum information. Recently, an elegant separability criterion to detect the entanglement of arbitrary-dimensional bipartite states is presented in Shi (2024 J. Phys. A: Math. Theor. 57 075302) by applying the positive operator valued measurements based on the equiangular tight frames (ETFs). Here we derive two enhanced separability criteria for detecting bipartite entanglement in arbitrary-dimensional quantum states using ETFs. Furthermore, we prove that they are not weaker than the criterion proposed in Shi (2024 J. Phys. A: Math. Theor. 57 075302).
Abstract Finite tight frames play an important role in miscellaneous areas, including quantum information theory. Here we apply a class of tight frames, equiangular tight frames, to address the problem of detecting the entanglement of bipartite states. Here we derive some entanglement criteria based on positive operator-valued measurements built from equiangular tight frames. We also present a class of entanglement witnesses based on the equiangular tight frames. At last, we generalize the entanglement criterion for bipartite systems to multipartite systems.
Abstract The detection of quantum entanglement represents a fundamental challenge in quantum measurement theory within the context of quantum information science. In this work, we propose a family of separability criteria based on mutually unbiased equiangular tight frames [IEEE Trans. Inf. Theory 67 , 1656–1667 (2021)], which constitute a natural generalization of both mutually unbiased bases and equiangular tight frames. The proposed criteria are applicable to a broad class of quantum systems, including bipartite systems with equal or unequal subsystem dimensions, as well as multipartite systems with heterogeneous dimensional components. In the examined cases, namely isotropic, Bell-diagonal, and Werner states, the proposed entanglement detection criteria demonstrate superior performance relative to several established methods. Crucially, these criteria rely solely on a small set of local measurements, which significantly enhances their experimental feasibility for the characterization of entanglement in unknown quantum states. Overall, the MUETF-based criteria provide more robust and versatile tools for entanglement verification across diverse quantum architectures.
Abstract Improvement of the detection ability of quantum entanglement is one of the essential tasks in quantum computing and quantum information. Finite tight frames play a fundamental role in a wide variety of areas and, generally, each application requires a specific class of frames and is closely related to quantum measurement. It is worth noting that a maximal set of complex equiangular vectors is closely related to a symmetric informationally complete measurement. Hence, our goal in this work is to propose a series of separability criteria assigned to a finite tight frame and some well-known inequalities in different quantum systems, respectively. In addition, some tighter criteria to detect entanglement for many-body quantum states are presented in arbitrary dimensions. Finally, the effectiveness of the proposed entanglement detection criteria is illustrated through some detailed examples.
Abstract Methods of quantum information processing often appear in terms of specially selected states. For example, mutually unbiased bases (MUBs) and symmetric informationally complete measurements are widely applied. Finite frames have found use in many areas including quantum information. Due to its specific inner structure, a single equiangular tight frame (ETF) allows one to formulate criteria to detect non-classical correlations. This study aims to approach entanglement detection with the use of mutually unbiased ETFs. Such frames are an interesting generalization of widely recognized MUBs. It still uses rank-one operators, but the number of outcomes can exceed the dimensionality. Several approaches are considered including separability criteria and entanglement witnesses. Separability criteria for multipartite systems are finally obtained.
In quantum information theory, measurements are represented by positive, operator-valued measures. In this paper, we use the operators corresponding to generalized equiangular measurements to construct positive maps. Their positivity follows from the properties of index of coincidence for few equiangular tight frames. These maps give rise to entanglement witnesses, which include as special cases many important classes considered in the literature. Additionally, we introduce separability criteria based on the correlation matrix and analyze them for various types of measurements.