Entropic Proofs of Singleton Bounds for Quantum Error-Correcting Codes

• Published in: IEEE transactions on information theory Vol. 68; no. 6; pp. 3942 - 3950
• Main Authors: Grassl, Markus, Huber, Felix, Winter, Andreas
• Format: Journal Article
• Language: English
• Published: New York IEEE 01-06-2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Codes; Decoding; Encoding; Entropy; Error correcting codes; Error correction; Error correction & detection; Error correction codes; Perturbation; Quantum codes; Quantum entanglement; Quantum mechanics; Qubit; Robustness; singleton bound
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2022.3149291

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 <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> and code length <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula> for given minimum distance <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula> 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.