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  • Open Access

Decidabilities of local unitary equivalence for entanglement witnesses and states

Yi Shen1,* and Lin Chen2,†

  • 1School of Science, Jiangnan University, Wuxi, Jiangsu 214122, China
  • 2LMIB (Beihang University), Ministry of Education, and School of Mathematical Sciences, Beihang University, Beijing 100191, China

  • *Contact author: yishen@jiangnan.edu.cn
  • †Contact author: linchen@buaa.edu.cn
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Phys. Rev. Research 7, 033258 – Published 16 September, 2025

DOI: https://doi.org/10.1103/bzfp-wxbt

Abstract

The problem of determining whether two states are equivalent by local unitary (LU) operations is important for quantum information processing. In this paper, we propose an alternative perspective to study this problem by comparing the decidabilities of LU equivalence (also known as LU decidabilities for short) between entanglement witnesses and states. We introduce a relation between sets of Hermitian operators in terms of the LU decidability. Then, we compare the LU decidability for the set of entanglement witnesses to the LU decidabilities for several sets of states. By comparison, we establish a hierarchy of these sets in terms of LU decidabilities. Moreover, we realize that the simultaneous LU (SLU) equivalence between tuples of mutually orthogonal projectors is crucial to LU equivalent operators. We reveal by examples that for two tuples of projectors, the partial SLU equivalence cannot ensure the overall SLU equivalence. Generally, we present a necessary and sufficient condition such that two tuples of states are SLU equivalent.

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Phys. Rev. Research 7, 033258– Published 16 September, 2025
  • Received 10 April 2025
  • Accepted 29 August 2025
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