通用量子编码
Universal quantum coding
- Freie Universität Berlin（柏林自由大学）
- Scuola Normale Superiore（比萨高等师范学院）
- Università degli Studi di Salerno（萨莱诺大学）
- INFN, Sezione di Napoli（意大利国家核物理研究所那不勒斯分部）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
通过Schur-Weyl对偶性建立通用量子纠错原理，实现无需先验知识的纠缠提取与量子通信，达到最优速率与样本复杂度。
AI中文摘要：
从噪声二分态中提取纠缠以及通过噪声信道进行可靠的量子通信是量子信息理论中的基本任务，然而最优编码方案通常依赖于对底层状态或信道的先验知识。在此，通过Schur-Weyl对偶性，我们建立了一个完全消除这种依赖性的通用量子纠错原理。我们表明，由未知二分混合态的多个相同副本的局部Schur-Weyl分解产生的不可约置换空间构成最大纠缠的量子编码空间，其错误结构完全由表示论决定。纠正这些对称性分辨的错误，可以得到用于纠缠提取和量子通信的通用协议，这些协议以指数衰减的误差和最优的二阶修正实现相干信息速率。这些协议在局部幺正变换下不变，并允许精确恢复两个输入边缘态。在有限块长度下，其可达速率由相关的Schur-Weyl信息谱控制，我们将其与底层状态的Petz-Rényi条件熵联系起来。由此产生的样本复杂度仅取决于相关的谱秩，而非各自的希尔伯特空间维度。特别地，对于全局秩为O(poly n)的n量子比特态，我们证明Petz-Rényi条件熵承诺足以实现最优的样本高效纠缠提取。总之，这些结果将Schur-Weyl对偶性确定为最优、免层析的通用量子编码的基本且通用的机制，将纠缠提取、量子纠错和量子通信置于统一的表示论框架内。
英文摘要：
Entanglement distillation from noisy bipartite states and reliable quantum communication over noisy channels are fundamental tasks in quantum information theory, yet optimal coding schemes typically rely on prior knowledge of the underlying state or channel. Here, through Schur-Weyl duality, we establish a universal quantum error-correction principle that removes this dependence entirely. We show that the irreducible permutation spaces arising from the local Schur-Weyl decomposition of many identical copies of an unknown bipartite mixed state form maximally entangled quantum code spaces, with an error structure determined entirely by representation theory. Correcting these symmetry-resolved errors yields universal protocols for entanglement distillation and quantum communication that achieve coherent-information rates with exponentially vanishing error and optimal second-order correction. The protocols are invariant under local unitaries and allow both input marginals to be recovered exactly. At finite block length, their achievable rates are governed by an associated Schur-Weyl information spectrum, which we relate to Petz-Rényi conditional entropies of the underlying state. The resulting sample complexity depends only on the relevant spectral ranks, rather than on the respective Hilbert space dimensions. In particular, for $n$-qubit states of $O(\operatorname{poly} n)$ global rank, we show that a Petz-Rényi conditional-entropy promise suffices for optimal sample-efficient entanglement distillation. Altogether, these results identify Schur-Weyl duality as a fundamental and general mechanism for optimal, tomography-free universal quantum coding, placing entanglement distillation, quantum error correction and quantum communication within a unified representation-theoretic framework.
