Measuring unitary invariants with the quantum switch

Bargmann invariants, multivariate traces of states, completely characterize any unitary-invariant property of a set of states. Unitary invariants enable the description of quantum resources such as basis-independent coherence and imaginarity, nonstabilizerness, and contextuality. We show that the quantum switch, a higher-order process featuring indefinite causal order, can be used to measure Bargmann invariants of arbitrary order. We also show how simple Hadamard test circuits can deterministically simulate an arbitrary unitary quantum switch. Our results establish a solid bridge between the theory and applications of unitary invariants and higher-order maps in quantum mechanics.