A Minimax Converse for Quantum Channel Coding

TL;DR

This paper establishes a minimax converse bound for quantum channel coding assisted by positive partial transpose channels, enabling explicit finite blocklength analysis and linking quantum channel bounds to classical counterparts.

Contribution

It introduces a semidefinite programming-based minimax converse bound with saddle point properties for quantum channels, simplifying finite blocklength analysis and connecting quantum and classical channel bounds.

Findings

Finite blocklength bounds for quantum erasure channels mirror classical results.

Bounds for dephasing and depolarizing channels align with classical binary symmetric channel bounds.

The convex bound simplifies optimization via channel symmetries.

Abstract

We prove a one-shot "minimax" converse bound for quantum channel coding assisted by positive partial transpose channels between sender and receiver. The bound is similar in spirit to the converse by Polyanskiy, Poor, and Verdu [IEEE Trans. Info. Theory 56, 2307-2359 (2010)] for classical channel coding, and also enjoys the saddle point property enabling the order of optimizations to be interchanged. Equivalently, the bound can be formulated as a semidefinite program satisfying strong duality. The convex nature of the bound implies channel symmetries can substantially simplify the optimization, enabling us to explicitly compute the finite blocklength behavior for several simple qubit channels. In particular, we find that finite blocklength converse statements for the classical erasure channel apply to the assisted quantum erasure channel, while bounds for the classical binary symmetric channel apply to both the assisted dephasing and depolarizing channels. This implies that these qubit channels inherit statements regarding the asymptotic limit of large blocklength, such as the strong converse or second-order converse rates, from their classical counterparts. Moreover, for the dephasing channel, the finite blocklength bounds are as tight as those for the classical binary symmetric channel, since coding for classical phase errors yields equivalently-performing unassisted quantum codes.