发表机构
Bocconi University; University of Copenhagen; University of British Columbia; LMU Munich & MCQST（博科尼大学; 哥本哈根大学; 不列颠哥伦比亚大学; 慕尼黑大学与量子科学与技术中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对 Aaronson 基于 Schrödinger 方案的隐变量理论，构造反例证明其鲁棒性猜想不成立，并提出保留其他性质的修改版本以恢复鲁棒性。
AI 中文摘要
受量子基础与复杂性理论启发，Aaronson 将 Schrödinger 提出的一个方案形式化为一种隐变量理论。对于任意量子态与酉算子，该 Schrödinger 理论通过 Sinkhorn 算法赋予一个联合概率分布：对酉算子逐项取模后，缩放其列与行，使得边际分布分别匹配初始与最终量子态的 Born 规则。他猜想该映射是鲁棒的，即输入中逆多项式小的扰动会导致联合分布逆多项式小的扰动，这对复杂性理论应用至关重要。我们对此猜想给出了一个反例：我们构造一个纯态和两个指数接近的酉算子，但 Schrödinger 理论赋予的联合概率分布在至少一个条目上相差至少一个逆线性项，且在总变差距离上相差一个常数。我们还提出了 Schrödinger 理论的一个修改版本，该版本满足鲁棒性，同时保留其所有其他理想性质。这完整刻画了隐变量理论可同时满足的 Aaronson 公理集合。
英文摘要
Motivated by quantum foundations and complexity theory, Aaronson formalized a hidden-variable theory inspired by a proposal by Schrödinger. To any quantum state and unitary, this Schrödinger theory assigns a joint probability distribution via the Sinkhorn algorithm: rescale the columns and rows of the entrywise modulus of the unitary so that the marginals match the Born rule for the initial and final quantum states, respectively. He conjectured that this map is robust, i.e., inverse polynomially small perturbations in the inputs lead to inverse polynomially small perturbations of the joint distribution, which is important for complexity theoretic applications. We present a counterexample to this conjecture: We construct a pure state and two unitaries that are exponentially close, yet the joint probability distributions assigned by the Schrödinger theory differ by at least an inverse-linear term in at least one entry, and by a constant in total variation distance. We also propose a modified version of Schrödinger's theory that satisfies robustness, while retaining all of its other desirable properties. This yields a complete picture of which axioms of Aaronson can be simultaneously satisfied by hidden-variable theories.
Comments22 pages, 1 table
