Entropic Proofs of Singleton Bounds for Quantum Error-Correcting Codes
Entropic Proofs of Singleton Bounds for Quantum Error-Correcting Codes
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DOI：
10.1109/tit.2022.3149291
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发表时间：
2020-10
影响因子：
2.5
通讯作者：
M. Grassl;Felix Huber;A. Winter
中科院分区：
文献类型：
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作者：
M. Grassl;Felix Huber;A. Winter
我们证明了使用von Neumann熵不等式的一个相对简单的推理给出了量子纠错码(QECC)的量子Singleton界的一个健壮的证明。对于纠缠辅助量子纠错码(EAQECC)和催化码(CQECC)，一类广义量子Singleton界[Brun等人，IEEE学报.信息理论60(6)：3073-3089(2014)]被认为成立了很多年，直到最近我们中的一个人发现了反例[MG，Phys。修订A 103,020601(2021年)]。在这里，我们通过证明正确的广义量子单态界来纠正这种情况，推广了上述量子纠错码的证明方法；我们还证明了EAQECC的纠缠-通信权衡的信息理论紧界。对于给定的最小距离$d$，所有的界都与块长度$n$和码长$k$有关，我们证明了它们是健壮的，因为它们对于仅纠正小于$d$个字母的大部分擦除错误的码具有小的扰动。与经典情况相比，根据最小距离是小于还是大于块长度的一半，边界呈现出定性不同的形式。我们还提供了一个传播规则：任何纯QECC产生的EAQECC具有相同的距离和维度，但具有更短的块长度。
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.