Asymptotic distinguishability of Haar-averaged measurement models

Abstract

We study discrimination problems generated by the same basic Haar-random measurement mechanism at two observational levels. First, we derive an explicit expression for the type-II error in the task of discriminating a Haar-random measure-and-prepare channel from the identity channel $\mathbb{I}$, using a coherence-sensitive entangled tester. Second, after passing to the induced classical measurement records, we compare two random measurement models: one induced by a single collective unitary of the form $U^{\otimes (n_1+n_2)}$ with $U\in U(d)$, and another induced by independent local unitaries $U_1^{\otimes n_1}\otimes U_2^{\otimes n_2}$. For the associated Haar-averaged aggregate histogram laws, in which the block of origin of each count is not retained, we obtain closed-form formulas and quantify their discrepancy through the total variation distance. We derive asymptotic expressions in the fixed-$N$, large-$d$ regime, the fixed-$d$, large-$N$ regime, the sparse joint-scaling regime $N=o(\sqrt d)$, and the critical scaling regime $N/\sqrt d\to c$. We also identify the block-resolved pair-of-histograms law, showing that the aggregate total variation distance is a coarse-grained lower bound on the distinguishability available when block labels are retained.