Universal classical-quantum channel resolvability and private channel coding
Takaya Matsuura, Masahito Hayashi, Min-Hsiu Hsieh
TL;DR
The paper tackles fully universal private channel coding for classical-quantum channels by introducing an explicit codebook-property—a Schreier-graph spectral gap—that guarantees universal channel resolvability. It develops universal resolvability coding via radical spectral expanders and ties it to induced representations to obtain channel-independent constructions. It then combines this resolvability framework with universal c-q coding based on Schur-Weyl duality to realize a fully universal private channel coding protocol that attains the known optimal rate. The results illuminate a novel connection between graph expanders, representation theory, and secure quantum communication, and they yield a channel-dimension-agnostic resolvability guarantee with broad applicability.
Abstract
We address the problem of constructing fully universal private channel coding protocols for classical-quantum (c-q) channels. Previous work constructed universal decoding strategies, but the encoder relied on random coding, which prevents fully universal code construction. In this work, we resolve this gap by identifying an explicit structural property of codebooks -- namely, the spectral expansion of an associated Schreier graph -- that guarantees universal channel resolvability. Our analysis reveals how this property can be related to channel resolvability through the theory of induced representation. Our main technical result shows that when the transition matrix of a graph associated with a codebook has a large spectral gap, the channel output induced by uniformly sampled codewords is asymptotically indistinguishable from the target output distribution, independently of the channel. This establishes the first deterministic, channel-independent construction of resolvability codebooks. Building on this, we construct a fully universal private channel coding protocol by combining it with universal c-q channel coding based on the Schur-Weyl duality. With appropriate modifications to the requirements on codebooks of universal c-q channel coding, we show that our fully universal private channel coding achieves the known optimal rate. This work thus sheds new light on the expander property of a graph in the context of secure communication.