An Approximation Algorithm for Approximation Rank

One of the strongest techniques available for showing lower bounds on bounded-error communication complexity is the logarithm of the approximation rank of the communication matrix---the minimum rank of a matrix which is entrywise close to the communication matrix. Krause showed that the logarithm of approximation rank is a lower bound in the randomized case, and later Buhrman and de Wolf showed it could also be used for quantum communication complexity. As a lower bound technique, approximation rank has two main drawbacks: it is difficult to compute, and it is not known to lower bound the model of quantum communication complexity with entanglement. We give a polynomial time constant factor approximation algorithm to the logarithm of approximation rank, and show that the logarithm of approximation rank is a lower bound on quantum communication complexity with entanglement.