Strong direct product theorems for quantum communication and query complexity
Proceedings of the forty-third annual ACM symposium on Theory of computing
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Consider the following {\em Simultaneous Message Passing} ($\smp$) model for computing a relation $f \subseteq \cX \times \cY \times \cZ$. In this model $\alice$, on input $x \in \cX$ and $\bob$, on input $y\in\cY$, send one message each to a third party $\referee$ who then outputs a $z \in \cZ$ such that $(x,y,z)\in f$. We first show optimal {\em Direct sum} results for all relations $f$ in this model, both in the quantum and classical settings, in the situation where we allow shared resources (shared entanglement in quantum protocols and public coins in classical protocols) between $\alice$ and $\referee$ and $\bob$ and $\referee$ and no shared resource between $\alice$ and $\bob$. This implies that, in this model, the communication required to compute $k$ simultaneous instances of $f$, with constant success overall, is at least $k$-times the communication required to compute one instance with constant success. This in particular implies an earlier Direct sum result, shown by Chakrabarti, Shi, Wirth and Yao~\cite{ChakrabartiSWY01} for the Equality function (and a class of other so-called robust functions), in the classical $\smp$ model with no shared resources between any parties. Furthermore we investigate the gap between the $\smp$ model and the one-way model in communication complexity and exhibit a partial function that is exponentially more expensive in the former if quantum communication with entanglement is allowed, compared to the latter even in the deterministic case.