Discriminating idempotent quantum channels

TL;DR

This paper investigates the asymptotic discrimination of idempotent quantum channels, establishing conditions for divergence collapse, error exponents, and the strong converse property, with applications to GNS-symmetric channels.

Contribution

It provides a complete characterization of asymptotic discrimination behavior for idempotent quantum channels under certain conditions, including strong converse results.

Findings

Divergences collapse to a single-letter expression under a shared invariant state.

All error exponents are explicitly computable with no adaptive advantage.

Discrimination rates for GNS-symmetric channels converge exponentially fast.

Abstract

We study binary discrimination of idempotent quantum channels. When the two channels share a common full-rank invariant state, we show that a simple image inclusion condition completely determines the asymptotic behavior: when it holds, a broad family of channel divergences collapse to a closed-form, single-letter expression, regularization is unnecessary, and all error exponents (Stein/Chernoff/strong-converse) are explicitly computable with no adaptive advantage. Crucially, this yields the strong converse property for this channel family, which is an important open problem for general channels. When the inclusion fails, asymmetric exponents become infinite, implying perfect asymptotic discrimination. We apply the results to GNS-symmetric channels, showing discrimination rates for large number of self iterations converge exponentially fast to those of the corresponding idempotent peripheral projections. If the two channels do not share a common invariant state, we provide a single-letter converse bound on the regularized sandwiched R\'{e}nyi cb-divergence, which suffices to establish a strong converse upper bound on the Stein exponents.