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.
翻译:我们显示,使用冯纽曼的相对简单的推理,可以有力地证明Singronton的量子定值符合量子误差校正代码(QECC)。对于缠绕式辅助量子误差校正代码(EAQECC)和催化代码(CQECC),一种通用量子单质定值[Brun等人,IEEETrans.Inf.Theory 60(6)/3073-3089(2014)],我们认为,在多年中一直坚持,直到最近,我们中的一个人发现了一个反比例[MG,Phys.Rev. A 103,020601(2021)]。在这里,我们通过证明正确的通用量定值单质差错校准码(EAQECC)和催化代码(C),扩展上述证据法;我们还证明,Enational-theal-comnational Tradeferoff(EEQEC), 信息-Terral-strate bounds the made list leas mind of leastlearal legal destrations), 我们显示它们保持了更短的距离, 或更小的准确的缩缩缩。