A low-order Bargmann invariant hierarchy for set coherence

Abstract Set coherence is a basis-independent relational form of quantum coherence: a finite family of quantum states is set incoherent exactly when all its members are diagonal in one common basis. We determine how much low-order Bargmann data are needed to decide this property. For two states, second-order data are complete for qubits but fail for qutrits, while complete third-order data are sufficient for qutrits but fail already in dimension four. We then show that fourth-order, ordering-sensitive Bargmann invariants give the first universal pairwise criterion for set coherence. Applied to all unordered pairs, this criterion yields a complete test for arbitrary finite families. In dimensional four system, there exist a set of coherent states and a set of incoherent states with identical Bargmann invariants of order at most three. We also interpret the hierarchy as a noncommutative moment hierarchy and give a dimension adaptive low-copy quantum computational test for whether a family of states admits a common diagonal representation. The result provides a low-order hierarchy connecting cyclic trace invariants with the noncommutativity that prevents a common incoherent basis.