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International Journal of Quantum Information

Local unitary equivalence for diagonal bipartite quantum states

Yanjun Chu, Mengli Liu, Keke Zhang

Published May 16, 2025• 2 citations
Publisher

Abstract

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.

DOI
10.1142/s0219749925500145

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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.
Local unitary equivalence is an important ingredient for quantifying and classifying entanglement. Verifying whether or not two quantum states are local unitary equivalent is a crucial problem, where only the case of multipartite pure states is solved. For mixed states, however, the verification of local unitary equivalence is still a challenging problem. In this paper, based on the coefficient matrices of generalized Bloch representations of quantum states, we find a variety of local unitary invariants for arbitrary-dimensional bipartite quantum states. These invariants are operational and can be used as necessary conditions for verifying the local unitary equivalence of two quantum states. Furthermore, we extend the construction to the arbitrary-dimensional multipartite case. We finally apply these invariants to estimate concurrence, a vital entanglement measure, showing the practicability of local unitary invariants in characterizing entanglement.
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.
Local unitary equivalence is central to entanglement quantification and classification. Identifying the local unitary equivalence remains a formidable challenge. We address this problem for a class of quantum states with one highly degenerate eigenvalue and the rest non-degenerate simple eigenvalues that are pivotal to structured resources in quantum resource theory. We introduce a ``reduction" procedure that maps each state to a ``reduced state" by nullifying the highest-multiplicity eigenvalue and prove that the local unitary equivalence of the original states is equivalent to that of their reduced counterparts. For the resulting pure or non-degenerate reduced states, we employ the existing invariants or fixed-point subgroup criteria to establish a complete discrimination framework, although the existing criteria can not directly identify the local unitary equivalence of the original states. We also verify the local unitary equivalence of two families of single-parameterized multipartite mixed states constructed by perturbing absolutely maximally entangled states from distinct combinatorial origins, demonstrating the efficacy and generality of our approach.
Research on quantum states often focuses on the correlation between nonlocal effects and local unitary invariants, among which local unitary equivalence plays a significant role in quantum state classification and resource theories. This paper focuses on the local unitary equivalence of multipartite quantum states in quantum information theory, aiming to determine a complete set of invariants that identify their local unitary orbits; these invariants are crucial for deriving polynomial invariants and describing the physical properties preserved under local unitary transformations. The study deeply explores the characterization of local unitary equivalence and the method of detecting entanglement using local unitary Bargmann invariants. Taking two-qubit systems as an example, it verifies the measurability of invariants that determine equivalence and establishes a connection between Makhlin fundamental invariants (a complete set of 18 local unitary invariants for two-qubit states) and local unitary Bargmann invariants. These Bargmann invariants, related to the traces of products of density operators and marginal states, can be measured through cycle tests (an extended form of swap tests).
Local unitary (LU) equivalent states are crucial for understanding the structure of quantum entanglement. We investigate the LU equivalence based on the LU Bargmann invariants (LUBIs). We consider for which states the LUBIs are sufficient to ensure the LU equivalence. We focus on a class of multipartite states with complete local commutativity, called CLC states. We first show that for multipartite CLC states, all the LUBIs are sufficient to ensure the local permutation equivalence, if their single-party marginals are diagonal and non-degenerate. By virtue of the fact that the CLC property remains unchanged under local operations, we verify that the LUBIs are also sufficient to ensure the LU equivalence of multipartite CLC states whose single-party marginals are non-degenerate. Furthermore, when the single-party marginals are degenerate, we propose concrete examples of two-qudit and three-qubit systems, showing that there are states which share all the LUBIs but are not LU equivalent.
The correspondence between local unitary equivalence of bipartite quantum states and simultaneous orthogonal equivalence is thoroughly investigated and strengthened. It is proved that local unitary equivalence can be studied through simultaneous similarity under projective orthogonal transformations, and four parametrization independent algorithms are proposed to judge when two density matrices on ℂd1 ⊗ ℂd2 are locally unitary equivalent in connection with trace identities, Kronecker pencils, Albert determinants and Smith normal forms.