Finite-blocklength classical communication over the quantum erasure channel with and without classical feedback

2026-09-30

AI Summary

This study addresses the unresolved problem of determining the optimal success probability for classical communication over finite blocklength quantum erasure channels without entanglement assistance, as well as the impact of feedback mechanisms. By leveraging adaptive quantum coding, tetrahedral state product codes, and dimensional bound analysis—combined with the relative majorization property and hypothesis testing converse bounds—the authors rigorously derive optimal performance limits for both scenarios with and without feedback. The work provides exact closed-form expressions for the success probability, proves that feedback yields a strict capacity gap and performance gain, and quantifies the resulting difference in error exponents. Ultimately, this research establishes a theoretical foundation for code design and for evaluating the utility of feedback in classical communication over quantum channels under finite resource constraints.

Abstract

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.

Problem

• quantum erasure channel
• finite-blocklength communication
• classical feedback
• optimal success probability
• channel capacity

Innovation

• quantum erasure channel
• finite-blocklength communication
• classical feedback
• dimension bound
• relative majorization