发表机构
School of Electrical and Computer Engineering, Cornell University（康奈尔大学电气与计算机工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究了量子擦除信道在有无经典反馈下的有限块长经典通信，给出了最优成功概率的精确表达式，并证明了反馈在特定参数下能带来严格优势，同时提供了无反馈时的逆定理和错误指数。
AI 中文摘要
我们确定了在没有初始共享纠缠且有无噪声经典反馈辅助的情况下，通过有限次使用量子擦除信道传输固定数量经典消息的最优成功概率。对于输入维度$d$、擦除概率$p$、块长$n$以及$M$个等概率消息，最优成功概率等于$E[\min\{1,d^K/M\}]$，其中$K$是参数为$n$和$1-p$的二项分布随机变量。逆定理允许任意自适应量子编码器、量子存储器和接收端仪器。其主要成分是一个关于无噪声量子通信与经典反馈的初等维度界，通过固定经典控制而不以观察到的转录本为条件来证明。一种经典协议传输表示消息的整数的基$d$数字，重复每个数字直到接收端确认收到或达到指定块长，对于每个整数$M$都能达到该界。我们还建立了擦除信道输出的相对优超性质，并精确评估了一个假设检验逆定理，在没有反馈的情况下恢复了相同的数值界。该逆定理在没有反馈时不一定可达：四次使用量子比特擦除信道和四个消息给出了严格差距。然而，采用四面体量子比特态的产品态编码优于所有具有这些参数的经典二进制擦除码。我们给出了一个显式的消息大小公式，并证明了有界余数正态近似和平均成功强逆指数在没有反馈的情况下不变。我们还确定了低于容量的反馈辅助错误指数，证明在临界速率以上它与无反馈指数一致，并在较低速率下给出了无反馈界。
英文摘要
We determine the optimal success probability for transmitting a fixed number of classical messages through a finite number of uses of the quantum erasure channel, assisted by noiseless classical feedback and without initial shared entanglement. For an input dimension $d$, an erasure probability $p$, a blocklength $n$, and $M$ equiprobable messages, the optimal success probability is equal to $E[\min\{1,d^K/M\}]$, where $K$ is binomial with parameters $n$ and $1-p$. The converse allows arbitrary adaptive quantum encoders, quantum memories, and receiver instruments. Its main ingredient is an elementary dimension bound for noiseless quantum communication with classical feedback, proved by fixing the classical controls without conditioning the sender's state on the observed transcript. A classical protocol that transmits the base-$d$ digits of an integer representing the message, repeating each digit until the receiver acknowledges its reception or the prescribed blocklength is reached, attains the bound for every integer $M$. We also establish a relative-majorization property of erasure-channel outputs and exactly evaluate a hypothesis-testing converse, recovering the same numerical bound without feedback. That converse need not be achievable without feedback: four uses of the qubit erasure channel and four messages give a strict gap. A product-state code employing tetrahedral qubit states nevertheless outperforms every classical binary erasure code with these parameters. We give an explicit message-size formula and show that the bounded-remainder normal approximation and the average-success strong-converse exponent are unchanged without feedback. We also determine the feedback-assisted error exponent below capacity, prove that it agrees with the no-feedback exponent above a critical rate, and give no-feedback bounds at lower rates.
Comments46 pages, 11 figures
