Efficient protocols for generating bipartite classical distributions and quantum states

Abstract

We investigate the fundamental problem of generating bipartite classical distributions or quantum states. By designing efficient communication protocols and proving their optimality, we establish a number of intriguing connections to fundamental measures in optimization, convex geometry, and information theory.1. To generate a classical distribution P(x, y), we tightly characterize the minimum amount of quantum communication needed by the psd-rank of P (as a matrix), a measure recently proposed by Fiorini, Massar, Pokutta, Tiwary and de Wolf (Proceedings of the 44th A CM Symposium on Theory of Computing, pages 95-106, 2012) in studies of the minimum size of extended formulations of optimization problems such as TSP. This echos the previous characterization for the optimal classical communication cost by the nonnegative rank of P. The result is obtained via investigating the more general case of bipartite quantum state generation and designing an optimal protocol for it.2. When an approximation of e is allowed to generate a distribution (X, Y) ~ P, we present a classical protocol of the communication cost O((C(X, Y) + 1)/e), where C(X, Y) is common information, a well-studied measure in information theory introduced by Wyner (IEEE Transactions on Information Theory, 21(2):163-179, 1975). This also links nonnegative rank and common information, two seemingly unrelated quantities in different fields.3. For approximately generating a quantum pure state |ψ>, we completely characterize the minimum cost by a corresponding approximate rank, closing a possibly exponential gap left in Ambainis, Schulman, Ta-Shma, Vazirani and Wigderson (SIAM Journal on Computing, 32(6):1570-1585, 2003).