Discriminating idempotent quantum channels

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 Rényi cb-divergence, which suffices to establish a strong converse upper bound on the Stein exponents.

Figures (2)

• Figure 1: Illustration of a general parallel strategy with $n$ uses of the black-box channel, and a joint binary POVM measurement $\{M, \mathbbm{1} - M\}$ at the end. The strategy is fully specified by a joint input state $\nu$.
• Figure 2: Illustration of a general adaptive strategy $\Theta$ with $n$ uses of the black-box channel, and a joint binary POVM measurement $\{M, \mathbbm{1} - M\}$ at the end. The top row uses the given black-boxes while the bottom row depicts the memory system $R$. The strategy is fully specified by an input state $\nu$ and intermediate input-preparation channels $\Lambda_2, \ldots ,\Lambda_n$.

Theorems & Definitions (26)

• Definition 2.1
• Lemma 2.2: Wilde2020amortized
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Remark 3.1
• Theorem 3.2
• Corollary 3.3
• Lemma 3.4
• Remark 3.5