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arXiv:2503.06240 · Quantum Information Processing

Local unitary equivalence of tripartite quantum states in terms of trace identities

Isaac Dobes, Naihuan Jing

Published May 28, 2025• 0 citations
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Abstract

Abstract In this paper, we generalize the major results from Jing-Yang-Zhao’s paper "Local Unitary Equivalence of Quantum States and Simultaneous Orthogonal Equivalence," which established a correspondence between local unitary (LU) equivalence and simultaneous orthogonal (SO) equivalence of bipartite quantum states, and then used this correspondence to show that the problem of determining LU equivalence of bipartite states reduces to checking trace identities. In particular, we extend both results to tripartite quantum states. We are able to do this by utilizing a hypermatrix algebra framework and by applying a powerful generalization of Specht’s criterion proved in Futorny-Horn-Sergeichuk’s paper "Specht’s Criterion for Systems of Linear Mappings." With our established hypermatrix algebra framework and the aforementioned generalization of Specht’s criterion, it is apparent that our results can be extended to arbitrary multipartite quantum states, however there are some practical limitations which are explored and discussed towards the end.

DOI
10.1007/s11128-025-04784-9

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Abstract In this paper, we generalize the major results from Jing-Yang-Zhao’s paper "Local Unitary Equivalence of Quantum States and Simultaneous Orthogonal Equivalence," which established a correspondence between local unitary (LU) equivalence and simultaneous orthogonal (SO) equivalence of bipartite quantum states, and then used this correspondence to show that the problem of determining LU equivalence of bipartite states reduces to checking trace identities. In particular, we extend both results to tripartite quantum states. We are able to do this by utilizing a hypermatrix algebra framework and by applying a powerful generalization of Specht’s criterion proved in Futorny-Horn-Sergeichuk’s paper "Specht’s Criterion for Systems of Linear Mappings." With our established hypermatrix algebra framework and the aforementioned generalization of Specht’s criterion, it is apparent that our results can be extended to arbitrary multipartite quantum states, however there are some practical limitations which are explored and discussed towards the end.
In this paper we present a modified version of the proof given Jing-Yang-Zhao's paper titled "Local Unitary Equivalence of Quantum States and Simultaneous Orthogonal Equivalence," which established the correspondance between local unitary equivalence and simultaneous orthogonal equivalence of 222-qubits. Our modified proof utilizes a hypermatrix algebra framework, and through this framework we are able to generalize this correspondence to 333-qubits. Finally, we apply a generalization of Specht's criterion (first proved in "Specht's Criterion for Systems of Linear Mappings" by V. Futorney, R. A. Horn, and V. V. Sergeichuk) to reduce the problem of local unitary equivalence of 333-qubits to checking trace identities and a few other easy-to-check properties. We also note that all of these results can be extended to 222 and 333 qudits if we relax the notion of LU equivalence to quasi-LU equivalence, as defined in the aforementioned paper by Jing et. al.
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.
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.
Orthogonal arrays (OAs) are combinatorial configurations with applications in experimental design, error-correcting codes, and quantum information. Local unitary (LU) equivalence provides a natural framework for classifying multipartite entangled states. Using the correspondence between OAs and quantum states, Goyeneche and Życzkowski [Phys. Rev. A, 2014, 90: 022316] posed the problem of determining when OAs are LULULU equivalent. In this paper, we construct OAs from generator matrices and establish conditions for Fourier-based LULULU equivalence of OAs and irredundant orthogonal arrays (IrOAs). For prime alphabets, we identify the Fourier partners of the linear OAs considered here with the arrays of their corresponding dual codes. This gives an explicit connection between LULULU equivalence and coding theory. We also identify families of LULULU equivalent OAs arising from specific classes of linear codes.
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.
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.