Connes’ embedding problem and winning strategies for quantum XOR games
Connes’ embedding problem and winning strategies for quantum XOR games
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Connes 的量子异或游戏的嵌入问题和获胜策略
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发表时间：
2017
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通讯作者：
Samuel J. Harris
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作者：
Samuel J. Harris
我们认为量子异或游戏，定义在工作中的Regev和Vidick [ACM trans.理论7，43（2015）]，从哈里斯和保尔森的工作中定义的酉相关性的角度[Integr.方程式操作理论89，125（2017）]。我们证明了量子XOR博弈在张量积模型（分别为交换模型）中的获胜偏差等于其相关线性泛函在相应模型的酉相关集上的范数。我们证明了Connes嵌入问题有一个正解当且仅当每个量子异或博弈的纠缠偏差等于交换偏差。特别地，嵌入问题等价于确定在交换模型中具有获胜策略的每个量子XOR博弈G在近似有限维模型中是否也具有获胜策略。
We consider quantum XOR games, defined in the work of Regev and Vidick [ACM Trans. Comput. Theory 7, 43 (2015)], from the perspective of unitary correlations defined in the work of Harris and Paulsen [Integr. Equations Oper. Theory 89, 125 (2017)]. We show that the winning bias of a quantum XOR game in the tensor product model (respectively, the commuting model) is equal to the norm of its associated linear functional on the unitary correlation set from the appropriate model. We show that Connes’ embedding problem has a positive answer if and only if every quantum XOR game has entanglement bias equal to the commuting bias. In particular, the embedding problem is equivalent to determining whether every quantum XOR game G with a winning strategy in the commuting model also has a winning strategy in the approximate finite-dimensional model.