A lower bound for the bounded round quantum communication complexity of set disjointness

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/sub i/ /spl sube/ [n]. These parties communicate with each other by transmitting qubits to determine with high probability the value of some function F of their combined input (X/sub 1/,...,X/sub t/). We consider the class of Boolean valued functions whose value depends only on X/sub 1/ /spl cap/.../spl cap/ X/sub t/; that is, for each F in this class there is an f/sub F/ : 2/sup [n]/ /spl rarr/ {0,1}, such that F(X/sub 1/,...,X/sub t/) = f/sub F/(X/sub 1/ /spl cap/.../spl cap/ X/sub t/). We show that the t-party k-round communication complexity of F is /spl Omega/(s/sub m/(f/sub F/)/(k/sup 2/)), where s/sub m/(f/sub F/) stands for the monotone sensitivity of f/sub F/' and is defined by s/sub m/(f/sub F/) = /sup /spl utri// max/sub S/spl sube//[n] |{i : f/sub F/(S /spl cup/ {i}) /spl ne/ f/sub F/(S)}|. For two-party quantum communication protocols for the set disjointness problem, this implies that the two parties must exchange /spl Omega/(n/k/sup 2/) qubits. An upper bound of O(n/k) can be derived from the O(/spl radic/n) upper bound due to S. Aaronson and A. Ambainis (2003). For k = 1, our lower bound matches the /spl Omega/(n) lower bound observed by H. Buhrman and R. de Wolf (2001) (based on a result of A. Nayak (1999)), and for 2 /spl les/ k /spl Lt/ n/sup 1/4 /, improves the lower bound of /spl Omega/(/spl radic/n) shown by A. Razborov (2002). For protocols with no restrictions on the number of rounds, we can conclude that the two parties must exchange /spl Omega/(n/sup 1/3/) qubits. This, however, falls short of the optimal /spl Omega/ (/spl radic/n) lower bound shown by A. Razborov (2002). Our result is obtained by adapting to the quantum setting the elegant information-theoretic arguments of Z. Bar-Yossef et al. (2002). Using this method we can show similar lower bounds for the L/sub /spl infin// function considered in Z. Bar-Yossef et al. (2002).