arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2001.04383quant-phcs.CCmath.OA
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MIP *= RE

MIP*=RE

Tsinghua University（清华大学）；Massachusetts Institute of Technology（麻省理工学院）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；University of California, Berkeley（加州大学伯克利分校）；Columbia University（哥伦比亚大学）

Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, Henry Yuen

备注：
250 pages. v4: Final version, accepted to Annals of Mathematics. Multiple corrections. Added link to Lean formalization of main result
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AI总结：

该研究证明MIP*类等于RE类，整合多项测试与框架得出推论，反驳了Connes嵌入猜想并解答Tsirelson问题。

摘要：

我们证明，由经典验证者与多个共享纠缠的全能量子证明者交互可判定的语言类MIP*，等于递归可枚举语言类RE。我们的证明基于Natarajan和Vidick（FOCS 2018）的量子低度测试、Ji等人（2020）的经典个体低度测试，整合了Natarajan和Wright（FOCS 2019）的最新进展，并结合了Fitzsimons等人（STOC 2019）的递归压缩框架。该结果的一个直接推论是，存在从停机问题到判定两人非局部游戏的纠缠值为1或至多1/2的问题的有效归约。利用已知关联，纠缠值的不可判定性意味着对Tsirelson问题的否定回答：我们通过提供一个明确例子证明，量子张量积关联集的闭包C_qa严格包含于量子交换关联集C_qc。继Fritz（Rev. Math. Phys. 2012）和Junge等人（J. Math. Phys. 2011）的工作之后，我们的结果反驳了冯·诺依曼代数理论中的Connes嵌入猜想。

英文摘要：

We show that the class MIP* of languages that can be decided by a classical verifier interacting with multiple all-powerful quantum provers sharing entanglement is equal to the class RE of recursively enumerable languages. Our proof builds upon the quantum low-degree test of (Natarajan and Vidick, FOCS 2018) and the classical low-individual degree test of (Ji, et al., 2020) by integrating recent developments from (Natarajan and Wright, FOCS 2019) and combining them with the recursive compression framework of (Fitzsimons et al., STOC 2019). An immediate byproduct of our result is that there is an efficient reduction from the Halting Problem to the problem of deciding whether a two-player nonlocal game has entangled value $1$ or at most $1/2$. Using a known connection, undecidability of the entangled value implies a negative answer to Tsirelson's problem: we show, by providing an explicit example, that the closure $C_{qa}$ of the set of quantum tensor product correlations is strictly included in the set $C_{qc}$ of quantum commuting correlations. Following work of (Fritz, Rev. Math. Phys. 2012) and (Junge et al., J. Math. Phys. 2011) our results provide a refutation of Connes' embedding conjecture from the theory of von Neumann algebras.

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