Fooling One-Sided Quantum Protocols

We use the venerable "fooling set" method to prove new lower bounds on the quantum communication complexity of various functions. Let f : X x Y -> {0,1} be a Boolean function, fool^1(f) its maximal fooling set size among 1-inputs, Q_1^*(f) its one-sided-error quantum communication complexity with prior entanglement, and NQ(f) its nondeterministic quantum communication complexity (without prior entanglement; this model is trivial with shared randomness or entanglement). Our main results are the following, where logs are to base 2: - If the maximal fooling set is "upper triangular" (which is for instance the case for the equality, disjointness, and greater-than functions), then we have Q_1^*(f) >= 1/2 log fool^1(f) - 1/2, which (by superdense coding) is essentially optimal for functions like equality, disjointness, and greater-than. No super-constant lower bound for equality seems to follow from earlier techniques. - For all f we have Q_1^*(f) >= 1/4 log fool^1(f) - 1/2. - NQ(f) >= 1/2 log fool^1(f) + 1. We do not know if the factor 1/2 is needed in this result, but it cannot be replaced by 1: we give an example where NQ(f) \approx 0.613 log fool^1(f).