Lower Bounds in Communication Complexity Based on Factorization Norms

We introduce a new method to derive lower bounds oil randomized and quantum communication complexity. Our method is based oil factorization norms, a notion from Banach Space theory. This approach gives its access to several powerful tools from this area such as normed spaces duality and Grothendiek's inequality. This extends the arsenal of methods for deriving lower bounds in communication complexity. As we show, our method subsumes most of the previously known general approaches to lower bounds on communication complexity. Moreover, we extend all (but one) of these lower bounds to the realm of quantum communication complexity with entanglement. Our results also shed some light on the question how much communication can be saved by using entanglement. It is known that entanglement call save one of every two qubits, and examples for which this is tight are also known. It follows from our results that this bound oil the saving in communication is tight almost always.