Singleton Bounds for Entanglement-Assisted Classical and Quantum Error Correcting Codes

TL;DR

This paper derives Singleton bounds for entanglement-assisted classical and quantum error-correcting codes using quantum Shannon theory, revealing the achievable rate regions and their geometric structure.

Contribution

It introduces a quantum Shannon theoretic approach to establish bounds and characterize the rate region for entanglement-assisted hybrid codes, including their geometric properties.

Findings

The rate region is contained within a polytope derived from a memoryless erasure channel.

Most of the rate region is attainable with sufficiently large local alphabet size.

All but one extremal line segment of the rate region are shown to be attainable.

Abstract

We show that entirely quantum Shannon theoretic methods, based on von Neumann entropies and their properties, can be used to derive Singleton bounds on the performance of entanglement-assisted hybrid classical-quantum (EACQ) error correcting codes. Concretely, we show that the triple-rate region of qubits, cbits and ebits of possible EACQ codes over arbitrary alphabet sizes is contained in the quantum Shannon theoretic rate region of an associated memoryless erasure channel, which turns out to be a polytope. We show that a large part of this region is attainable by certain EACQ codes, whenever the local alphabet size (i.e. Hilbert space dimension) is large enough, in keeping with known facts about classical and quantum minimum distance separable (MDS) codes: in particular, all of its extreme points and all but one of its extremal lines. The attainability of the remaining one extremal line segment is left as an open question.