Entropic proofs of Singleton bounds for quantum error-correcting codes

TL;DR

This paper presents entropic proofs of the quantum Singleton bounds for various quantum error-correcting codes, correcting previous misconceptions and establishing tight bounds on code parameters with robustness considerations.

Contribution

It provides a simple entropy-based proof of the quantum Singleton bound and corrects the generalized bound for entanglement-assisted codes, including tight bounds on entanglement-communication tradeoffs.

Findings

Proved the quantum Singleton bound using von Neumann entropy inequalities.

Corrected the generalized quantum Singleton bound for EAQECC and CQECC.

Established robust bounds that tolerate small perturbations in error correction.

Abstract

We show that a relatively simple reasoning using von Neumann entropy inequalities yields a robust proof of the quantum Singleton bound for quantum error-correcting codes (QECC). For entanglement-assisted quantum error-correcting codes (EAQECC) and catalytic codes (CQECC), a type of generalized quantum Singleton bound [Brun et al., IEEE Trans. Inf. Theory 60(6):3073--3089 (2014)] was believed to hold for many years until recently one of us found a counterexample [MG, Phys. Rev. A 103, 020601 (2021)]. Here, we rectify this state of affairs by proving the correct generalized quantum Singleton bound, extending the above-mentioned proof method for QECC; we also prove information-theoretically tight bounds on the entanglement-communication tradeoff for EAQECC. All of the bounds relate block length $n$ and code length $k$ for given minimum distance $d$ and we show that they are robust, in the sense that they hold with small perturbations for codes which only correct most of the erasure errors of less than $d$ letters. In contrast to the classical case, the bounds take on qualitatively different forms depending on whether the minimum distance is smaller or larger than half the block length. We also provide a propagation rule: any pure QECC yields an EAQECC with the same distance and dimension, but of shorter block length.