Simple communication complexity separation from quantum state antidistinguishability

Abstract

A set of n pure quantum states is called antidististinguishable if there exists an n-outcome measurement that never outputs the outcome 'k' on the kth quantum state. We describe sets of quantum states for which any subset of three states is antidistinguishable and use this to produce a two-player communication task that can be solved with logd qubits, but requires one-way communication of at least log(4/3)(d-1)-1≈0.415(d-1)-1 classical bits. The advantages of the approach are that the proof is simple and self-contained-not needing, for example, to rely on hard-to-establish prior results in combinatorics- A nd that with slight modifications, nontrivial bounds can be established in any dimension ≥3. The task can be framed in terms of the separated parties solving a relation. We show, however, that for this particular task, the separation disappears if two-way classical communication is allowed, or if the task need only be solved with bounded error. Finally, we state a conjecture regarding antidistinguishability of sets of states, and provide some supporting numerical evidence. If the conjecture holds, then there is a two-player communication task that can be solved with logd qubits, but requires exact one-way communication of ω(dlogd) classical bits.