Infinite Dimensional Asymmetric Quantum Channel Discrimination

TL;DR

This paper extends quantum channel discrimination theory to infinite dimensions, establishing key lemmas, showing asymptotic equivalence of strategies, and providing explicit bounds and examples.

Contribution

It proves quantum Stein's lemma for infinite-dimensional channels, demonstrates the additivity of geometric R\'enyi divergence, and constructs explicit bounds relating adaptive and parallel strategies.

Findings

Adaptive strategies offer no asymptotic advantage over parallel ones under certain conditions.

The geometric R\'enyi divergence satisfies a chain rule and is additive for channels in infinite dimensions.

Explicit bounds are provided for the difference in discrimination errors between strategies.

Abstract

We study asymmetric binary channel discrimination, for qantum channels acting on separable Hilbert spaces. We establish quantum Stein's lemma for channels for both adaptive and parallel strategies, and show that under finiteness of the geometric R\'enyi divergence between the two channels for some $\alpha > 1$, adaptive strategies offer no asymptotic advantage over parallel ones. One major step in our argument is to demonstrate that the geometric R\'enyi divergence satisfies a chain rule and is additive for channels also in infinite dimensions. These results may be of independent interest. Furthermore, we not only show asymptotic equivalence of parallel and adaptive strategies, but explicitly construct a parallel strategy which approximates a given adaptive $n$-shot strategy, and give an explicit bound on the difference between the discrimination errors for these two strategies. This extends the finite dimensional result from [B. Bergh et al., arxiv:2206.08350]. Finally, this also allows us to conclude, that the chain rule for the Umegaki relative entropy in infinite dimensions, recently shown in [O. Fawzi, L. Gao, and M. Rahaman, arxiv:2212.14700v2] given finiteness of the max divergence between the two channels, also holds under the weaker condition of finiteness of the geometric R\'enyi divergence. We give explicit examples of channels which show that these two finiteness conditions are not equivalent.