带噪声纠缠的非局域游戏与通信复杂性

中文摘要 AI 辅助

我们研究噪声对量子非局域性和纠缠辅助通信复杂性理论的影响，考虑Alice和Bob可共享任意多对带噪声EPR对的模型下的非局域游戏与纠缠辅助通信复杂性，分析四种噪声模型：去极化噪声、幺正噪声、偏置重置噪声和删除噪声。研究结果如下：1. 针对所有这些噪声模型，在不对策略所用测量做任何假设的情况下，给出CHSH游戏值关于噪声参数的上界；2. 证明除偏置重置噪声外，一般非局域游戏的并行重复定理，还证明独特游戏的改进并行重复定理，在非平凡噪声区域下，该并行重复率小于CHSH的量子并行重复率；3. 利用独特游戏并行重复定理及CHSH游戏定义的关系，证明带噪声与无噪声纠缠的通信复杂性之间存在分离，这意味着在同一非平凡噪声区域下，从带噪声EPR对蒸馏n个EPR对需要Ω(n)的双向通信下界；4. 证明任何交互式纠缠辅助通信协议均可由带噪声共享随机数的SMP通信协议模拟，通信量呈指数级膨胀，该结论推广了用带噪声共享随机数模拟无噪声共享随机数的已知结果；5. 证明在所有四种噪声模型下，计算具有恒定通信量的Equality函数所需带噪声EPR对副本数的多项式下界，此前已有匹配的上界结果，且该研究回答了文献中关于是否需要对数级带噪声共享比特即可实现通信的开放问题。

英文摘要

We study the impact of noise on the theories of quantum nonlocality and entanglement-assisted communication complexity. We consider non-local games and entanglement-assisted communication complexity in a model where Alice and Bob may share arbitrarily many noisy EPR pairs. We study four noise models: depolarizing noise, unital noise, biased reset noise, and erasure noise. Our results are as follows: 1. We upper bound the value of the CHSH game under all these noise models in terms of the noise parameter, without any assumptions on the measurements used in the strategy. 2. We prove a parallel repetition theorem for general non-local games under all noise models except biased reset noise; we prove an improved parallel repetition theorem for unique games. Our parallel repetition rate is smaller than the quantum parallel repetition rate for CHSH in a nontrivial noise regime. 3. Using our unique-game parallel repetition theorem and a relation defined by the CHSH game, we prove a separation between communication complexity with noisy vs noiseless entanglement. This implies an $Ω(n)$ two-way communication lower bound for distilling $n$ EPR pairs from noisy EPR pairs, in the same nontrivial noise regime. 4. We show that any interactive entanglement-assisted communication protocol can be simulated by an SMP communication protocol with noisy shared randomness, with an exponential blowup in communication. This generalizes a known result on the simulation of noiseless shared randomness with noisy shared randomness. 5. We show a polynomial lower bound on the number of copies of noisy EPR pairs required to compute the Equality function with constant communication, under all four noise models. The previous result gives a matching upper bound, and moreover, this answers an open question in the literature on whether logarithmically many noisy shared bits suffice for communication.

发表机构

• Quantum Computing Research Centre（量子计算研究中心）
• Hon Hai (Foxconn) Research Institute（鸿海（富士康）研究院）
• Institute for Quantum Computing（量子计算研究所）
• University of Waterloo（滑铁卢大学）

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