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Local Unitary Equivalence of Tripartite Quantum States In Terms of Trace Identities

Isaac Dobes, Naihuan Jing

TL;DR

The paper tackles LU equivalence for tripartite and, more generally, multipartite quantum states by recasting density matrices as hypermatrix representations and linking LU to simultaneous orthogonal (and quasi-LU) equivalence via a generalized Specht criterion. The approach uses hypermatrix operations, mode unfoldings, and the geometric hyperdeterminant to derive invariants and to reduce LU testing to trace identities involving structured words in the T-tensors, aided by partial-trace reductions and quiver-based generalizations. For bipartite and tripartite states, the authors provide concrete criteria: norm equalities, sign conditions, and determinant-type invariants that together imply LU (or quasi-LU) equivalence when trace identities hold up to computed bounds. They likewise outline extensions to arbitrary multipartite systems, noting significant computational and technical challenges, but highlighting a scalable framework that can complement existing methods (e.g., Li–Qiao) for special state classes. The work thus advances practical, trace-identity–based tests for LU classification in multipartite quantum systems, with explicit caveats and potential for broader applicability in entanglement characterization.

Abstract

In this paper we present a modified version of the proof given Jing-Yang-Zhao's paper "Local Unitary Equivalence of Quantum States and Simultaneous Orthogonal Equivalence," which established the correspondence between local unitary (LU) equivalence and simultaneous orthogonal equivalence of bipartite quantum states. Our modified proof utilizes a hypermatrix algebra framework, and with this framework we are able to generalize this correspondence to tripartite quantum states. Finally, we apply a generalization of Specht's criterion proved in Futorny-Horn-Sergeichuk' paper "Specht's Criterion for Systems of Linear Mappings" to \textit{essentially} reduce the problem of local unitary equivalence of tripartite quantum states to checking trace identities and a few other LU invariants. We also note that all of these results can be extended to arbitrary multipartite quantum states, however there are some practical limitations.

Local Unitary Equivalence of Tripartite Quantum States In Terms of Trace Identities

TL;DR

The paper tackles LU equivalence for tripartite and, more generally, multipartite quantum states by recasting density matrices as hypermatrix representations and linking LU to simultaneous orthogonal (and quasi-LU) equivalence via a generalized Specht criterion. The approach uses hypermatrix operations, mode unfoldings, and the geometric hyperdeterminant to derive invariants and to reduce LU testing to trace identities involving structured words in the T-tensors, aided by partial-trace reductions and quiver-based generalizations. For bipartite and tripartite states, the authors provide concrete criteria: norm equalities, sign conditions, and determinant-type invariants that together imply LU (or quasi-LU) equivalence when trace identities hold up to computed bounds. They likewise outline extensions to arbitrary multipartite systems, noting significant computational and technical challenges, but highlighting a scalable framework that can complement existing methods (e.g., Li–Qiao) for special state classes. The work thus advances practical, trace-identity–based tests for LU classification in multipartite quantum systems, with explicit caveats and potential for broader applicability in entanglement characterization.

Abstract

In this paper we present a modified version of the proof given Jing-Yang-Zhao's paper "Local Unitary Equivalence of Quantum States and Simultaneous Orthogonal Equivalence," which established the correspondence between local unitary (LU) equivalence and simultaneous orthogonal equivalence of bipartite quantum states. Our modified proof utilizes a hypermatrix algebra framework, and with this framework we are able to generalize this correspondence to tripartite quantum states. Finally, we apply a generalization of Specht's criterion proved in Futorny-Horn-Sergeichuk' paper "Specht's Criterion for Systems of Linear Mappings" to \textit{essentially} reduce the problem of local unitary equivalence of tripartite quantum states to checking trace identities and a few other LU invariants. We also note that all of these results can be extended to arbitrary multipartite quantum states, however there are some practical limitations.

Paper Structure

This paper contains 15 sections, 9 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hypermatrix Algebra
  4. Cayley's Second Hyperdeterminant
  5. Quasi-LU Equivalence
  6. Bipartite States
  7. SO Equivalence and Trace Identities: Bipartite States
  8. Tripartite States
  9. SO-Equivalence and Trace Identities: Tripartite States
  10. Applying the Partial Trace and Futorny's Theorem
  11. Extending to Arbitrary Multipartite States
  12. Conclusion
  13. Appendix
  14. Specht's Criterion and Generalizations
  15. Quiver Representations and Futorny's Theorem

Key Result

Proposition 1

lim2013tensors Let $A\in F^{n_1\times...\times n_d}$ and suppose $\mathrm{Det}(A)$ exists. Then for any $X_1\in GL(n_1)$,..., $X_d\in GL(n_d)$, we have that where $m$ is the degree of $\mathrm{Det}(A)$. Consequently, $\mathrm{Det}$ is invariant under the multilinear matrix multiplication of $(X_1,...,X_d)$ with $X_i\in SL(n_i)$ (and in particular, $X_i\in SU(n_i)$) for each $i$.

Theorems & Definitions (11)

  • Proposition 1
  • Theorem 1: quasi-LU Equivalence and SO Equivalence: Bipartite States
  • proof
  • Theorem 2: Characterization of quasi-LU Equivalence: Bipartite States
  • Theorem 3: quasi-LU Equivalence and SO Equivalence: Tripartite States
  • proof
  • Lemma 1
  • Theorem 4: Characterizing quasi-LU Equivalence
  • Proposition 2: Specht's criterion (real version)
  • Theorem 5
  • ...and 1 more