基于随机与确定性置换的量子信道经典通信超加性

中文摘要 AI 辅助

自Hastings证明量子信道上经典通信的超加性以来，大量研究致力于为这一最初由Haar随机幺正矩阵的测度集中性确立的现象寻找结构性解释。本研究的核心发现是，Haar随机性可被随机置换取代，且不会改变导致非加性的极限几何结构。这种替换将幺正矩阵上的连续问题转化为0-1置换矩阵上的离散组合问题，从而为去随机化开辟了路径。Bordenave与Collins的定理表明随机置换具有所需的极限行为，而O'Donnell和Wu的算法则提供了一种确定性渐近构造，当信道参数和精度固定时，其运行时间为规模的多项式时间。因此，随机构造可在渐近算法意义上去随机化，尽管寻找简单的闭式或实际可计算的反例仍未解决。最后，Chen、Garza-Vargas、Tropp及van Handel给出了定量的随机置换估计：存在作用于大小为N≤5.422×10^116216的集合上的57836025个置换构成的元组，使得相关有限维信道表现出非加性。这一巨大数值仍是实际构造的障碍。

英文摘要

Since Hastings' proof of superadditivity of classical communication over quantum channels, considerable effort has been devoted to finding a structural explanation of this phenomenon that was originally established by concentration of measure for Haar random unitaries. The main observation of this work is that Haar randomness can be replaced by random permutations without changing the limiting geometry responsible for nonadditivity. This replacement turns a continuous problem over unitary matrices into a discrete combinatorial problem over zero--one permutation matrices, and thereby opens a path toward derandomization. The theorem of Bordenave and Collins shows that random permutations have the required limiting behavior and the algorithm of O'Donnell and Wu then provides a deterministic asymptotic construction, running in polynomial time in the size when the channel parameters and accuracy are fixed. Thus the random construction can be derandomized in an asymptotic algorithmic sense. Finally, a quantitative random permutation estimate by Chen, Garza-Vargas, Tropp and van Handel gives a fully numerical estimate: there exists a tuple of 57,836,025 permutations acting on a set of size \[ N \le 5.422\times 10^{116216}\] such that the associated finite dimensional channel exhibits nonadditivity. This enormous value remains an obstacle to a practical construction.

发表机构

• Concordia University（康考迪亚大学）
• Syracuse University（雪城大学）
• University of Waterloo（滑铁卢大学）

机构由 AI 辅助整理，请以论文原文为准。