Bounded-error quantum state identification and exponential separations in communication complexity

We consider the problem of bounded-error quantum state identification: given either state 0 or state 1, we are required to output `0', `1' or `?' ("don't know"), such that conditioned on outputting `0' or `1', our guess is correct with high probability. The goal is to maximize the probability of not outputting `?'. We prove a direct product theorem: if we are given two such problems, with optimal probabilities a and b, respectively, and the states in the first problem are pure, then the optimal probability for the joint bounded-error state identification problem is O(ab). Our proof is based on semidefinite programming duality. Using this result, we present two exponential separations in the simultaneous message passing model of communication complexity. First, we describe a relation that can be computed with O(log n) classical bits of communication in the presence of shared randomness, but needs (n1/3 ) communication if the parties don'...

• Type: Conference
• Year: 2006
• Where: STOC
• Authors: Dmitry Gavinsky, Julia Kempe, Oded Regev, Ronald de Wolf
• Keywords: Algorithms, Bar-Yossef Et Al, CR/02 20040 Grants, French Research Ministry, STOC 2006