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Bargmann-invariant framework for local unitary equivalence and entanglement

Lin Zhang, Bing Xie, Yuanhong Tao

TL;DR

This paper develops a Bargmann invariant–based framework to classify multipartite quantum states under local unitary transformations and to detect entanglement. It defines LU Bargmann invariants $B_k$ and establishes a complete LU invariant set for two-qubit states, linking them to Makhlin invariants through generalized Schur-Weyl duality and showing measurability via cycle-test circuits. An explicit entanglement criterion is derived in terms of the 7 invariants required for two-qubit states, equivalent to the PPT condition, with discussion of extensions to higher dimensions. The work highlights experimental feasibility, offers a path toward efficient LU-based state discrimination, and suggests future directions in connecting Bargmann invariants to broader representations and random measurement frameworks.

Abstract

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).

Bargmann-invariant framework for local unitary equivalence and entanglement

TL;DR

This paper develops a Bargmann invariant–based framework to classify multipartite quantum states under local unitary transformations and to detect entanglement. It defines LU Bargmann invariants and establishes a complete LU invariant set for two-qubit states, linking them to Makhlin invariants through generalized Schur-Weyl duality and showing measurability via cycle-test circuits. An explicit entanglement criterion is derived in terms of the 7 invariants required for two-qubit states, equivalent to the PPT condition, with discussion of extensions to higher dimensions. The work highlights experimental feasibility, offers a path toward efficient LU-based state discrimination, and suggests future directions in connecting Bargmann invariants to broader representations and random measurement frameworks.

Abstract

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).

Paper Structure

This paper contains 21 sections, 33 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Bargmann invariant of a tuple of quantum states
  3. Local unitary equivalence of multipartite states
  4. Entanglement criterion via LU Bargmann invariants
  5. Discussion
  6. Conclusion
  7. Proof of Proposition \ref{['th:LUequiv']}
  8. Invariant theory
  9. The generalized Schur-Weyl duality
  10. Proof of Proposition \ref{['th:LUequiv']}
  11. Proof of Theorem \ref{['th:Bargmann-generators']}
  12. Product formula for two-qubit observables
  13. Product formula
  14. Auxiliary results
  15. Recurrence relation for the matrix power
  16. ...and 6 more sections

Key Result

Proposition 1

For any two $N$-partite states $\rho$ and $\sigma$ acting on $\mathbb{C}^{d_1}\otimes\cdots\otimes \mathbb{C}^{d_N}$, they are LU equivalent, i.e., $\sigma=g\rho g^\dagger$ for some $g\in \mathsf{LU}(\boldsymbol{d})$ for $\boldsymbol{d}=(d_1,\ldots,d_N)$, if and only if, for arbitrary positive integ where the meaning of $\mathbf{P}_{\boldsymbol{d},n}(\boldsymbol{\pi})$ will be explained immediatel

Theorems & Definitions (60)

  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Proposition 2: Vrana2012
  • Proposition 3
  • Theorem 3: Schur, Zhang2024
  • Theorem 4: The generalized Schur-Weyl duality, Grassl1998Turner2017
  • proof : Proof of Proposition \ref{['th:LUequiv']}
  • Definition 1
  • Proposition 4
  • ...and 50 more