关于光滑矩形界的一个强直积定理
A strong direct product theorem in terms of the smooth rectangle bound
摘要：
强直积定理指出，为了解决问题的k个实例，如果我们提供的资源少于计算一个实例所需的k倍，那么总体成功的概率在k中呈指数级小。在本文中，我们考虑双向公共硬币通信复杂性的模型，并证明了一个强直积定理的所有关系的光滑矩形界，介绍了由Jain和Klauck作为一般的下界方法在这个模型中。因此，我们的结果一致地暗示了一个强直积定理，对于所有的关系，可以使用平滑矩形边界来提供（渐进地）最优下界，例如内积、大于、集不相交性，差距-Hamming距离等。我们的结果还暗示了几个重要函数和关系的近似最优直积结果，这些函数和关系用于显示经典和量子通信之间的指数分离复杂度，使用矩形边界提供接近最优的下限，例如Raz [1999]，Gavinsky [2008]和Klartag和Regev [2011]。事实上，我们不知道任何关系，它是已知的，光滑的矩形边界不提供一个最佳的下限。这个下界包含了许多其他的下界方法，例如矩形界（又称腐败界）、光滑差异界（又称γ_2界），后者又包含了差异界、子分布界和条件最小熵界。我们使用信息理论的参数显示我们的结果。我们使用的一个关键工具是Braverman [2012]的采样协议，实际上是Kerenoval，Laplante，Lerays，Roland和Xiao [2012]使用的修改。
英文摘要：
A strong direct product theorem states that, in order to solve k instances of a problem, if we provide less than k times the resource required to compute one instance, then the probability of overall success is exponentially small in k. In this paper, we consider the model of two-way public-coin communication complexity and show a strong direct product theorem for all relations in terms of the smooth rectangle bound, introduced by Jain and Klauck as a generic lower bound method in this model. Our result therefore uniformly implies a strong direct product theorem for all relations for which an (asymptotically) optimal lower bound can be provided using the smooth rectangle bound, for example Inner Product, Greater-Than, Set-Disjointness, Gap-Hamming Distance etc. Our result also implies near optimal direct product results for several important functions and relations used to show exponential separations between classical and quantum communication complexity, for which near optimal lower bounds are provided using the rectangle bound, for example by Raz [1999], Gavinsky [2008] and Klartag and Regev [2011]. In fact we are not aware of any relation for which it is known that the smooth rectangle bound does not provide an optimal lower bound. This lower bound subsumes many of the other lower bound methods, for example the rectangle bound (a.k.a the corruption bound), the smooth discrepancy bound (a.k.a the γ_2 bound) which in turn subsumes the discrepancy bound, the subdistribution bound and the conditional min-entropy bound. We show our result using information theoretic arguments. A key tool we use is a sampling protocol due to Braverman [2012], in fact a modification of it used by Kerenidis, Laplante, Lerays, Roland and Xiao [2012].
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