Error exponent of activated non-signaling assisted classical-quantum channel coding

TL;DR

This work characterizes the error exponent for activated non-signaling assisted coding over classical-quantum channels, showing it matches the sphere packing bound and can be written as a single-letter optimization over Petz-Rényi divergences: $E^{ANS}(r)=\sup_{\alpha\in(0,1]} \frac{1-\alpha}{\alpha}\big(C_{\alpha}(\mathcal{W})-r\big)$. The authors establish both achievability and converse via SDP duality and a dual Petz-Rényi representation using Young inequalities, and prove that this bound holds for all rates in the non-trivial interval $(C_0(\mathcal{W}),C(\mathcal{W}))$, with activation enabling the bound. They show activation is unnecessary when zero-error NS capacity is positive or when outputs are pure states, and derive analogous results for fully quantum channels using Choi matrices. The results provide an operational interpretation of Petz Rényi divergences in the quantum setting and open directions for rounding techniques that translate NS bounds into plain, shared randomness, or shared entanglement strategies. Overall, the paper extends classical sphere packing insights to activated non-signaling assisted quantum scenarios and highlights pathways for fully quantum sphere packing and related coding reductions.

Abstract

We provide a tight asymptotic characterization of the error exponent for classical-quantum channel coding assisted by activated non-signaling correlations. Namely, we find that the optimal exponent--also called reliability function--is equal to the well-known sphere packing bound, which can be written as a single-letter formula optimized over Petz-Rényi divergences. Remarkably, there is no critical rate and as such our characterization remains tight for arbitrarily low rates below the capacity. On the achievability side, we further extend our results to fully quantum channels. Our proofs rely on semi-definite program duality and a dual representation of the Petz-Rényi divergences via Young inequalities.

Theorems & Definitions (34)

• Proposition 3.1
• proof : Proof of Proposition \ref{['prop-dual']}
• Theorem 4.1
• Proposition 4.2
• proof : Proof of Proposition \ref{['prop-exp-ach']}
• Proposition 4.3
• Lemma 4.4
• proof
• Proposition 5.1
• proof