Comparison of unknown unitary channels with multiple queries

Abstract

Comparison of quantum objects is the task of determining relational properties, such as whether two unknown objects are the same or different, without identifying the objects themselves. Multiple copies of unknown states or multiple queries to unknown channels are natural resources for improving comparison, and the optimal strategy for pure-state comparison with multiple copies is known. For unitary-channel comparison with multiple queries, however, the optimal strategy has been unclear because different queries can be arranged in parallel, sequentially, adaptively, or in more general causal structures. We study comparison of two unknown $d$-dimensional unitary channels using $N_1$ and $N_2$ queries to the two channels, respectively, under the promise that they are either identical Haar-random unitaries or independent Haar-random unitaries. We optimize over valid general testers, a class containing ordinary quantum testers, classically controlled causal orders, and indefinite-causal-order processes. We characterize the optimal minimum-error and one-sided unambiguous comparison probabilities for arbitrary finite query numbers. The upper bound in fact holds for a larger positive relaxation, while the optimum is attained by a valid parallel tester. The optimum is determined by a finite representation-theoretic parameter built from Young diagrams and Littlewood--Richardson coefficients. For fixed $N_1$, the performance is saturated for every $N_2\geq(d-1)N_1$. By contrast, the corresponding pure-state comparison parameter has no finite-$N_2$ saturation and reaches its known-reference limit only asymptotically, yielding continued success-probability improvement in the nontrivial decision regime. This highlights a sharp difference between comparison tasks for states and channels, analogous to distinctions known in quantum discrimination.