Commutativity from a single Bargmann invariant equality

TL;DR

This paper establishes a universal criterion for quantum state commutativity using Bargmann invariants, enabling practical tests that do not require full state tomography and have applications in quantum measurement and photonics.

Contribution

It provides a necessary and sufficient condition for quantum state commutativity based on measurable invariants, simplifying analysis in quantum information.

Findings

Derived a universal condition for pairwise commutativity of quantum states.

Simplified the identity for qubits to polynomial relations involving purities and overlaps.

Proposed practical commutativity tests that bypass full state tomography.

Abstract

Noncommutativity of states and observables is a fundamental signature of quantum theory, and a minimal requirement for nonclassicality. We provide a universal necessary and sufficient condition for pairwise commutativity of quantum states $\rho_1$ and $\rho_2$: they commute if and only if $\mathrm{tr}(\rho_1^2\rho_2^2) = \mathrm{tr}(\rho_1 \rho_2 \rho_1 \rho_2)$. For qubits the identity simplifies to an equality between polynomials of purities and of the two-state overlap $\mathrm{tr}(\rho_1\rho_2)$. These multivariate traces (known as Bargmann invariants) are directly measurable, allowing commutativity tests that bypass full state tomography. We point out possible applications to the analysis of POVM simulability and partial photonic distinguishability.