Anti-distinguishability of quantum channels

Abstract

We study single-query perfect anti-distinguishability/exclusion tasks of finite quantum channel ensembles, primarily using associated Choi operators and process POVMs. Weak perfect anti-distinguishability requires each measurement outcome to exclude an ensemble candidate with certainty, whereas the strong form additionally requires every candidate to have an exclusion outcome occurring with non-zero probability. For pairs of channels, both coincide with perfect distinguishability, whereas larger ensembles exhibit separations. We derive Choi-operator-support-based obstructions: weak exclusion requires at least two rank-deficient Choi operators, while strong exclusion requires the support of every candidate Choi operator to be strictly contained in the joint Choi support. The common Choi support must also lie in the kernel of the PPOVM normalisation. For commuting Choi operators, this condition is sufficient for weak exclusion when the normalisation also commutes with them. Conversely, a nonzero admixture of the same noise channel into every candidate precludes weak exclusion for any probe. We apply these results to measure-and-prepare, state-space contraction, Pauli, and replacer channels. In particular, the full permutation-channel ensemble of an unlabeled POVM is strongly anti-distinguishable exactly when some POVM effect is rank-deficient, while replacer-channel exclusion reduces to state exclusion. A Choi-commuting ensemble exhibits strong exclusion with an unassisted probe although no positive-definite PPOVM normalisation permits even weak exclusion. In contrast, commuting qubit phase gates, whose Choi operators need not commute, admit exact convex-hull criteria for weak and strong exclusion, both attainable without assistance.