Abstract
We show lower bounds in the multi-party quantum communication complexity model. In this model, there are t parties where the ith party has input X_i \subseteq \left[ n \right]. These parties communicate with each other by transmitting qubits to determine with high probability the value of some function F of their combined input (X1, . . .,Xt). We consider the class of boolean valued functions whose value depends only on X_1 \cap \cdots \cap X_t ; that is, for each F in this class there is an f_F :2^{\left[ n \right]} \to \{ 0,1\}, such that F(X_{1, \ldots ,} X_t ) = f_F (X_1 n \cdots nX_t ). We show that the t-party k-round communication complexity of F is \Omega (s_m (f_F )/(k^2 )), where s_m (f_F ) stands for the ‘monotone sensitivity of f_F’ and is defined by s_m (f_F ) \triangleq \max _{S \subseteq \left[ n \right]} \left| {\{ i:f_F } \right.(S \cup \{i\} ) \ne f_F (S)\left. \} \right|. For two-party quantum communication protocols for the set disjointness problem, this implies that the two parties must exchange \Omega (n/k^2 ) qubits. An upper bound of O(n/k) can be derived from the O(\sqrt n) upper bound due to Aaronson and Ambainis [AA03]. For k = 1, our lower bound matches the \Omega(n) lower bound observed by Buhrman and de Wolf [BdW01] (based on a result of Nayak [Nay99]), and for 2 \leqslant k \ll n^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}}, improves the lower bound of \Omega (\sqrt n) shown by Razborov [Raz02]. For protocols with no restrictions on the number of rounds, we can conclude that the two parties must exchange \Omega (n^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}) qubits. This, however, falls short of the optimal \Omega (\sqrt n) lower bound shown by Razborov [Raz02]. Our result is obtained by adapting to the quantum setting the elegant information-theoretic arguments of Bar-Yossef, Jayram, Kumar and Sivakumar [BJKS02b]. Using this method we can show similar lower bounds for the L\infty function considered in [BJKS02b].