证书博弈及其对经典对手界的推论
Certificate Games and Consequences for the Classical Adversary Bound
AI总结：
本文提出证书博弈复杂度，通过零通信博弈刻画经典对手界，并证明量子证书博弈复杂度可比量子查询复杂度大平方倍，最终揭示除私有随机性外所有证书博弈模型均坍缩到经典对手界。
摘要：
我们引入并研究了证书博弈复杂度，这是一种基于博弈获胜概率的复杂度度量。在该博弈中，两名玩家分别获得具有不同函数值的输入，并被要求在零通信设定下输出某个索引 $i$，使得 $x_i\neq y_i$。我们研究了证书博弈的四种版本，即私有随机、公共随机、共享纠缠和非信号博弈。证书博弈的公共随机变体给出了经典对手界的一个新刻画，经典对手界是随机化查询复杂度的一个下界，它作为量子（非负）量子对手界的经典版本被引入。我们证明，公共随机模型中的复杂度（因此也是经典对手界）被证书复杂度、期望证书复杂度和破坏复杂度所上界。另一方面，它被分数证书复杂度和随机化证书复杂度所下界。量子度量揭示了经典查询模型与量子查询模型之间一个有趣且令人惊讶的差异：量子证书博弈复杂度可以比量子查询复杂度大平方倍。我们使用非信号这一来自量子信息的概念，对 OR 函数的量子证书博弈复杂度给出了 $n$ 的下界，而该函数的量子查询复杂度为 $Θ(\sqrt{n})$，随后进一步证明这一“非信号瓶颈”适用于所有具有高敏感度、块敏感度、分数块敏感度以及经典对手界的函数。这意味着除私有随机性外，所有证书博弈模型都坍缩到经典对手界。我们考虑了证书博弈的单比特版本，其中两名玩家的输入被限制为汉明距离为 1，并给出了敏感度和谱敏感度的新刻画。
英文摘要：
We introduce and study Certificate Game complexity, a measure of complexity based on the probability of winning a game where two players are given inputs with different function values and are asked to output some index $i$ such that $x_i\neq y_i$, in a zero-communication setting. We study four versions of certificate games, namely private coin, public coin, shared entanglement and non-signaling games. The public-coin variant of certificate games gives a new characterization of the classical adversary bound, a lower bound on randomized query complexity which was introduced as a classical version of the quantum (non-negative) quantum adversary bound. We show that complexity in the public coin model (therefore also the classical adversary) is bounded above by certificate complexity, as well as by expectational certificate complexity and sabotage complexity. On the other hand, it is bounded below by fractional and randomized certificate complexity. The quantum measure reveals an interesting and surprising difference between classical and quantum query models: the quantum certificate game complexity can be quadratically larger than quantum query complexity. We use non-signaling, a notion from quantum information, to give a lower bound of $n$ on the quantum certificate game complexity of the OR function, whose quantum query complexity is $Θ(\sqrt{n})$, then go on to show that this ``non-signaling bottleneck'' applies to all functions with high sensitivity, block sensitivity, fractional block sensitivity, as well as classical adversary. This implies the collapse of all models of certificate games, except private randomness, to the classical adversary bound. We consider the single-bit version of certificate games, where the inputs of the two players are restricted to having Hamming distance 1, and give a new characterization of sensitivity and spectral sensitivity.
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