A direct product theorem for one-way quantum communication

We prove a direct product theorem for the one-way entanglement-assisted quantum communication complexity of a general relation f⊆X×Y×Z. For any 0 < ε < δ < 1/2 and any k ≥ 1, we show that [EQUATION] where Q1ε(f) represents the one-way entanglement-assisted quantum communication complexity of f with worst-case error ε and fk denotes k parallel instances of f. As far as we are aware, this is the first direct product theorem for the quantum communication complexity of a general relation - direct sum theorems were previously known for one-way quantum protocols for general relations, while direct product theorems were only known for special cases. Our techniques are inspired by the parallel repetition theorems for the entangled value of two-player non-local games, under product distributions due to Jain, Pereszlényi and Yao [24], and under anchored distributions due to Bavarian, Vidick and Yuen [4], as well as message compression for quantum protocols due to Jain, Radhakrishnan and Sen [29]. In particular, we show that a direct product theorem holds for the distributional one-way quantum communication complexity of f under any distribution q on X × Y that is anchored on one side, i.e., there exists a y* such that q(y*) is constant and q(x\y*) = q(x) for all x. This allows us to show a direct product theorem for general distributions, since for any relation f and any distribution p on its inputs, we can define a modified relation [EQUATION] which has an anchored distribution q close to p, such that a protocol that fails with probability at most ε for [EQUATION] under q can be used to give a protocol that fails with probability at most ε + ζ for f under p. Our techniques also work for entangled non-local games which have input distributions anchored on any one side, i.e., either there exists a y* as previously specified, or there exists an x* such that q(x*) is constant and q(y\x*) = q(y) for all y. In particular, we show that for any game G = (q,X×Y, A×B, V) where q is a distribution on X × Y anchored on any one side with constant anchoring probability, then [EQUATION] where ω* (G) represents the entangled value of the game G. This is a generalization of the result of [4], who proved a parallel repetition theorem for games anchored on both sides, i.e., where both a special x* and a special y* exist, and potentially a simplification of their proof.