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Journal of Physics A Mathematical and Theoretical

Schmidt number criterion via mutually unbiased equiangular tight frames

Lin qiao Che, Yi Long, Fan Wu, Liang Bao Tang

Published Oct 20, 2026• 0 citations

Abstract

Abstract With the rapid evolution of quantum information science, quantum entanglement remains a fundamental resource for quantum tasks. Nevertheless, the reliable and efficient detection of entanglement, especially the quantitative determination of the Schmidt number in high-dimensional quantum systems, continues to pose a significant and long-standing challenge. In this work, we introduce a novel Schmidt number criterion for bipartite quantum systems via mutually unbiased equiangular tight frames. Notably, this criterion unifies and substantially generalizes previous approaches that relied on mutually unbiased bases and equiangular measurements (Tavakoli and Morelli 2024 Phys. Rev. A 110 062417). Based on this framework, we also derive a lower bound on concurrence. Moreover, detailed examples demonstrate that our result outperforms several existing approaches. Consequently, this contribution advances both the foundational theory of entanglement quantification and the development of scalable, experimentally feasible tools for quantum information processing.

DOI: 10.1088/1751-8121/aeb695

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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.

Families of

k

-positive maps and Schmidt number witnesses from generalized equiangular measurements

Quantum entanglement is an important resource in many modern technologies, like quantum computation or quantum communication and information processing. Therefore, most interest is given to detect and quantify entangled states. Entanglement degree of bipartite mixed quantum states can be measured using the Schmidt number. Witnesses of the Schmidt number are closely related to

k

-positive linear maps, for which there is no general construction. Here, we use the generalized equiangular measurements to define a family of

k

-positive linear maps and the corresponding Schmidt number witnesses. We present examples of witnesses that detect two-parameter entangled states with Schmidt number

k+1

in any dimension. Our approach allows for a more efficient entanglement quantification than other known Schmidt number witnesses constructed from symmetric measurement operators.

Abstract Higher dimensional entangled states demonstrate significant advantages in quantum information processing tasks. The Schmidt number is a quantity of the entanglement dimension of a bipartite state. Here we build families of k-positive maps from the symmetric information complete positive operator-valued measurements and mutually unbiased bases, and we also present the Schmidt number witnesses, correspondingly. At last, based on the witnesses obtained from mutually unbiased bases, we show the distance between a bipartite state and the set of states with a Schmidt number less than k.