Bargmann-invariant framework for local unitary equivalence and entanglement

TL;DR

This paper introduces a Bargmann-invariant framework for analyzing local unitary equivalence and entanglement in quantum states, providing a method to identify invariants that classify states and detect entanglement.

Contribution

It develops a comprehensive set of invariants based on Bargmann invariants for classifying multipartite quantum states under local unitaries, linking them to known invariants and measurement techniques.

Findings

Verifies measurability of invariants for two-qubit states

Establishes connection between Bargmann invariants and Makhlin invariants

Proposes cycle tests for measuring invariants

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).