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 × Y → {0, 1} be a Boolean function, fool1(f) its maximal fooling set size among 1-inputs, Q1(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 Q1(f) ≥ 2 log fool 1(f)− 1 2 , which (by superdense coding) is essentially optimal for functions like equality, disjointness, and greaterthan. No super-constant lower bound for equality seems to follow from earlier techniques. For all f we have Q1(f) ≥ 4 log fool 1(f)− 1 2 . NQ(f) ≥ 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) ≈ 0.613 log fool1(f). 1998 ACM Subject Classification F.1.1 Models of Computation