Entropic Proofs of Singleton Bounds for Quantum Error-Correcting Codes

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.