Joint State-Channel Decoupling and One-Shot Quantum Coding Theorem

TL;DR

This paper introduces a joint state-channel decoupling method to derive a one-shot error exponent bound in quantum information, improving previous bounds and applying it to quantum channel coding with explicit exponential decay of error.

Contribution

It presents a novel joint decoupling approach that yields explicit error bounds in one-shot quantum information tasks, strengthening previous results.

Findings

Explicit error exponent bound expressed via sandwiched Rényi entropies

Improved bounds over previous decoupling results

Application to quantum channel coding with exponential error decay

Abstract

In this work, we consider decoupling a bipartite quantum state via a general quantum channel. We propose a joint state-channel decoupling approach to obtain a one-shot error exponent bound without smoothing, in which trace distance is used to measure how good the decoupling is. The established exponent is expressed in terms of a sum of two sandwiched R{\'e}nyi entropies, one quantifying the amount of initial correlation between the state and environment, while the other characterizing the effectiveness of the quantum channel. This gives an explicit exponential decay of the decoupling error in the whole achievable region, which was missing in the previous results [Commun. Math. Phys. 328, 2014]. Moreover, it strengthens the error exponent bound obtained in a recent work [IEEE Trans. Inf. Theory, 69(12), 2023], for exponent from the channel part. As an application, we establish a one-shot error exponent bound for quantum channel coding given by a sandwiched R\'enyi coherent information.