Commutativity from a single Bargmann invariant equality

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.