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 [16] as a generic lower bound method in this model. Our result therefore implies a strong direct product theorem for all relations for which an (asymptotically) optimal lower bound can be provided using the smooth rectangle bound. 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 known lower bound methods, for example the rectangle bound (a.k.a the corruption bound) [31], the smooth discrepancy bound (a.k.a the γ2 bound [28] which in turn subsumes the discrepancy bound), the subdistribution bound [17] and the conditional min-entropy bound [14] . As a consequence, our result reproves some of the known strong direct product results, for example for Inner Product [26] and Set-Disjointness [25, 14] . Our result also shows new strong direct product result for Gap-Hamming Distance [8, 33] and 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 [30], Gavinsky [11] and Klartag and Regev [32]. We show our result using information theoretic arguments. A key tool we use is a sampling protocol due to Braverman [5] , in fact a modification of it used by Kerenidis, Laplante, Lerays, Roland and Xiao [23].
Jain and Yao REF followed up with a strong direct product theorem for the communication complexity of all functions for which an optimal lower bound can be shown using the smooth rectangle bound.
A strong direct product theorem in terms of the smooth rectangle bound
ArXiv
ArXiv
Mathematics Computer Science