论文标题
量子界误差校正代码的单身界的熵证明
Entropic proofs of Singleton bounds for quantum error-correcting codes
论文作者
论文摘要
我们表明，使用von Neumann熵不等式的相对简单的推理产生了量子单胎绑定的量子误差校正代码（QECC）的强大证明。对于纠缠辅助量子误差校正代码（EAQECC）和催化代码（CQECC），一种类型的广义量子singleton结合[Brun等，IEEE，IEEE Trans。 inf。据信理论60（6）：3073--3089（2014）]直到最近我们中有人发现反例[mg，phys。修订版A 103，020601（2021）]。在这里，我们通过证明正确的Quantum Singleton绑定来纠正这种状况，从而扩展了QECC上述证明方法；我们还证明了有关EAQECC的纠缠沟通权衡的信息，从理论上讲，信息范围很紧。所有边界都将块长度$ n $和代码长度$ k $用于给定最小距离$ d $的代码长度$ k $，我们表明它们很健壮，从某种意义上说，它们对只有小于$ d $字母的大多数擦除错误的代码持有小扰动。与经典案例相反，界限在定性上不同的形式取决于最小距离是块长度的一半。我们还提供了一个传播规则：任何纯QECC都会产生具有相同距离和尺寸的EAQECC，但块长度较短。
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.
