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Physics Letters A

On local unitary equivalence and local permutation equivalence of bipartite quantum states under permutation transformations

Mengli Liu, Yanjun Chu, Keke Zhang, Yuhang Xie, Chenyang Cui

Published Dec 3, 2025• 1 citation
Publisher
DOI
10.1016/j.physleta.2025.131223

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Local unitary equivalence plays a crucial role in quantifying and categorizing the entanglement of quantum states. When it comes to two isospectral diagonal bipartite quantum states, determining whether they are locally unitary equivalent turns out to be a rather challenging problem. Intuitively speaking, we conjecture that for two diagonal quantum states, local unitary equivalence and local permutation equivalence between them are the same. By taking into account the cosets of the fixed point subgroup and the tensor decomposability of permutation matrices, we present the classifications of diagonal states in [Formula: see text]⊗[Formula: see text] and [Formula: see text]⊗[Formula: see text] under local permutation equivalence, respectively. Furthermore, relying on the classification results of diagonal states in [Formula: see text]⊗[Formula: see text] and [Formula: see text]⊗[Formula: see text], we demonstrate that if two diagonal quantum states are not locally permutation equivalent, then they cannot be locally unitary equivalent. This offers significant backing for the conjecture that two general bipartite diagonal quantum states are locally unitary equivalent if and only if they are locally permutation equivalent.
Permutation matrices are an old and widely used family of matrices in mathematics and physics. In this paper, we classify bipartite permutation matrices under local permutation equivalence when the equivalence is in the form of the Kronecker product of two permutation matrices. We classify all [Formula: see text] bipartite permutation matrices of Schmidt rank up to three. We also study the equivalence for general bipartite permutation gates in terms of Schmidt rank two. We apply our result to study the entangling power of two-qubit permutation gates. Under the local unitary equivalence, we show that the five distinct two-qubit gates under local permutation equivalence can be reduced to exactly three gates. Our results present a physical understanding of the permutation group from the point of view of quantum information and experiments.