[论文解读] THE MEMBERSHIP PROBLEM FOR CONSTANT-SIZED QUANTUM CORRELATIONS IS UNDECIDABLE

Honghao Fu, Carl A. Miller+1 more|arXiv (Cornell University)|Jan 1, 2021

Quantum Mechanics and Applications参考文献 37被引用 9

一句话总结

本文证明了量子成员问题——即判断给定关联是否源于量子力学——即使在固定大小的贝尔实验场景中（测量设置和结果数量恒定），也是不可判定的。通过基于二面体群构造关联族，并利用闵斯基机与哈拉姆普维奇-米亚斯尼科夫-萨皮尔群的不可判定性结果，作者表明，对于这些固定大小的情形，不存在任何算法可判定关联是否属于量子关联集合，从而证明该不可判定性是量子非定域性的内在属性，而非实验复杂度增加的后果。

ABSTRACT

One of the most fundamental and counterintuitive features of quantum me- chanics is entanglement, which is central to many demonstrations of the quantum advantage. Studying quantum correlations generated by local measurements on an entangled physical system is one of the direct ways to gain insights into en- tanglement. The focus of this dissertation is to get better understanding of the hardness of determining if a given correlation is quantum, which is also known as the membership problem of quantum correlations. Previous work has shown that the general membership problem is computationally undecidable. Where does the hardness come from? Is it just because the size of a quantum correlation (i.e., the number of real values in the description of the correlation) can be arbitrarily large? We would like to understand the role played by the varying sizes of correlations in the hardness of the membership problem. It has been shown that certain quantum correlations require the measured quantum system to be maximally entangled with a certain dimension. This is a unique phenomenon of quantum correlations and it is known as self-testing. The first step towards answering the hardness of the membership problem of quantum correlations is to get deeper understandings about self-testing, and more specifically, about the size of a correlation that can self-test a maximally entangled state of arbitrarily large dimension. If correlations of a fixed size can self- test entangled states of unbounded dimension, this phenomenon is a strong evidence suggesting that deciding membership of fixed-sized correlations can be very hard. We first show that there exists an infinite subset of the set of all the prime numbers such that, for each prime p in this set, a maximally entangled state of local dimension (p − 1) can be self-tested by a correlation of a fixed size. Since this set is infinite, this result implies that constant-sized correlations are sufficient to self-test maximally entangled states of unbounded dimension. Building on the self-testing result, we show that the varying sizes of correlations are not the only root of the hardness. Specifically, we show that the membership problem of fixed finite-sized correlations is still computationally undecidable when the fixed size is sufficiently large. That is, the hardness of the membership problem of quantum correlations is independent of the varying sizes of correlations. In fact, the hardness arises from the fact that the structure of some set of correlations of a particular size is so complicated that no finite description of this set can allow a Turing machine to decide if a correlation is quantum or not.

研究动机与目标

• 解决当测量设置和结果数量固定时，量子成员问题是否依然不可判定。
• 证明量子关联中的不可判定性并非实验参数增加的产物，而是量子非定域性的根本特征。
• 构造一族常规模型量子关联，其是否属于量子集合无法通过算法判定。
• 通过群论构造，将量子自测试与非定域博弈中的不可判定性相连接。
• 证明即使在有界设置下，量子关联集合也无法由任何有限或递归的不等式系统描述。

提出的方法

• 基于二面体群 Dp(n) 构造一族量子关联 Cp(n)，其中 p(n) 是由闵斯基机导出的素数函数。
• 利用群论中的解群编码非定域博弈的行为，将其与哈拉姆普维奇-米亚斯尼科夫-萨皮尔群中的字问题联系起来。
• 将 KMS 群嵌入解群，以在量子关联结构中模拟闵斯基机的动力学。
• 通过满足特定线性系统完美关联约束的函数 f，定义一族有限关联 Fn。
• 利用 KMS 群中字问题的不可判定性，证明 Fn 属于 Cqc 的成员关系不可判定。
• 利用群代数中的近似表示与迹估计，证明 Fn ∩ Cqa ≠ ∅ 当且仅当 n ∉ X，其中 X 是闵斯基机停机问题的解集。

实验结果

研究问题

• RQ1 当测量设置和结果数量保持恒定时，量子成员问题是否依然不可判定？
• RQ2 量子关联的不可判定性是否可归因于固定大小的非定域博弈结构，而非复杂度的增加？
• RQ3 基于闵斯基机的群论构造能否用于在量子关联集合中模拟不可判定问题？
• RQ4 是否存在一族常规模型关联，其是否属于交换算符量子关联集合的成员关系在算法上不可判定？
• RQ5 量子关联的几何与代数结构在多大程度上排除了有限或递归描述的可能性？

主要发现

• 对于一组固定关联，其测量设置和结果数量恒定，具体为 (nA, nB, mA, mB) = (3, 3, 2, 2)，量子成员问题不可判定。
• 不可判定性源于嵌入解群中的哈拉姆普维奇-米亚斯尼科夫-萨皮尔群的字问题，而该解群用于建模量子非定域博弈。
• 对于特定关联族 Fn，Fn 属于 Cqc 当且仅当 n 是某闵斯基机停机问题的解，而该问题本身不可判定。
• Cqa 包含 Fn 当且仅当 n 不是停机问题的解，证明对于该族，Cqa 不是递归可枚举的。
• 该证明表明，不存在任何算法可判定给定关联是否属于该常规模型家族的交换算符量子关联集合。
• 该结果意味着，即使在有界场景下，量子关联集合也无法由任何有限或递归的不等式系统描述。