发表机构
University of Illinois Urbana-Champaign; RWTH Aachen University; Forschungszentrum Jülich; Korea Institute for Advanced Study（伊利诺伊大学厄巴纳-香槟分校; 亚琛工业大学; 于利希研究中心; 韩国高等科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明稳定子码中Rényi相干信息量随Rényi指数非递减，并赋予其基于后选择匹配综合征的操作意义，同时上界普通相干信息量。
AI 中文摘要
Rényi相干信息量作为von Neumann相干信息量的可计算代理，被广泛用于研究混合态物质相以及有噪声量子纠错码中的可解码性转变。然而，由于它是两个Rényi熵之差，它不必在Rényi指数上单调，并且缺乏其von Neumann对应物的操作意义。在这里，我们针对稳定子码解决了这两个问题。首先，对于由独立Bernoulli事件产生的Pauli噪声，我们证明了Rényi-$n$相干信息量在$n \in \mathbb{Z}^+$上是非递减的。这源于一个一般性定理：如果独立的随机比特被线性映射到一个精细标签$T$和一个粗略标签$C$，那么Rényi熵差$H_n(C)-H_n(T)$在$n$上是非递减的。对于稳定子码，$T$是联合综合征-逻辑类，$C$是综合征，而该差值是Rényi-$n$相干信息量（相差一个常数）。同样的定理也适用于经典线性码和独立探测器错误模型。其次，对于任意随机Pauli噪声，我们通过在一个数据块和$n-1$个辅助块之间匹配综合征的后选择，赋予Rényi-$n$相干信息量一个操作意义。我们确定了这何时定义了一个量子信道，并表明Rényi-$n$相干信息量的饱和等价于后选择信道的渐近完美恢复。此外，Rényi-$n$相干信息量还上界了在任何综合征条件恢复后可达的普通相干信息量。
英文摘要
Rényi coherent information, a computable proxy for the von Neumann coherent information, is widely used to study mixed-state phases of matter and decodability transitions in noisy quantum error-correcting codes. However, being a difference of two Rényi entropies, it need not be monotonic in the Rényi index, and lacks the operational meaning of its von Neumann counterpart. Here we address both issues for stabilizer codes. First, for Pauli noise generated by independent Bernoulli events, we prove that the Rényi-$n$ coherent information is nondecreasing in $n \in \mathbb{Z}^+$. This follows from a general theorem: if independent random bits are mapped linearly to a fine label $T$ and a coarse label $C$, then the Rényi entropy difference $H_n(C)-H_n(T)$ is nondecreasing in $n$. For stabilizer codes, $T$ is the joint syndrome--logical class and $C$ is the syndrome, and the difference is the Rényi-$n$ coherent information up to a constant. The same theorem covers classical linear codes and independent detector error models. Second, for arbitrary stochastic Pauli noise, we give the Rényi-$n$ coherent information an operational meaning via postselection on matching syndromes between one data block and $n-1$ auxiliary blocks. We determine when this defines a quantum channel and show that saturation of the Rényi-$n$ coherent information is equivalent to asymptotically perfect recovery of the postselected channel. Moreover, the Rényi-$n$ coherent information also upper-bounds the ordinary coherent information achievable after any syndrome-conditioned recovery.
