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
ε,ζ>0 and any k≥1, we show that Q1−(1−ε)Ω(ζ6k/log∣Z∣)1(fk)=Ω(k(ζ5⋅Qε+12ζ1(f)−loglog(1/ζ))), where Qε1(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 quantum
communication. 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, Pereszl\'{e}nyi and Yao, and under anchored
distributions due to Bavarian, Vidick and Yuen, as well as message-compression
for quantum protocols due to Jain, Radhakrishnan and Sen.
Our techniques also work for entangled non-local games which have input
distributions anchored on any one side. 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 anchoring probability ζ, then ω∗(Gk)=(1−(1−ω∗(G))5)Ω(log(∣A∣⋅∣B∣)ζ2k) where ω∗(G)
represents the entangled value of the game G. This is a generalization of the
result of Bavarian, Vidick and Yuen, who proved a parallel repetition theorem
for games anchored on both sides, and potentially a simplification of their
proof.Comment: 31 pages, 1 figur