On quantum-classical equivalence for composed communication problems

An open problem in communication complexity proposed by several authors is to prove that for every Boolean function f, the task of computing f(x ∧ y) has polynomially related classical and quantum bounded-error complexities. We solve a variant of this question. For every f, we prove that the task of computing, on input x and y, both of the quantities f(x∧y) and f(x∨y) has polynomially related classical and quantum boundederror complexities. We further show that the quantum bounded-error complexity is polynomially related to the classical deterministic complexity and the block sensitivity of f. This result holds regardless of prior entanglement.