Efficient Protocols for Generating Bipartite Classical Distributions and Quantum States

• Published in: IEEE transactions on information theory Vol. 59; no. 8; pp. 5171 - 5178
• Main Authors: Jain, Rahul, Yaoyun Shi, Zhaohui Wei, Shengyu Zhang
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01-08-2013; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Access methods and protocols, osi model; Applied sciences; Approximation; Approximation methods; Communication complexity; Complexity theory; Correlation; Exact sciences and technology; Information theory; Information, signal and communications theory; Miscellaneous; Optimization; Probability distribution; probability distribution generation; Protocols; psd-rank; quantum correlation complexity; Quantum mechanics; Random variables; Signal and communications theory; Telecommunications; Telecommunications and information theory; Teleprocessing networks. Isdn; Measure theory; Probabilistic approach; quantum correlation complexity; Transmission protocol; Travelling salesman problem; probability distribution generation; Communication complexity; Probability distribution; psd-rank; Optimization; Quantum communication; Information theory
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2013.2258372

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 et al. (Proc. 44th ACM Symp. Theory Comput., pp. 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 ϵ is allowed to generate a distribution (X,Y)~P, we present a classical protocol of the communication cost O((C(X,Y)+1)/ϵ, where C(X,Y) is common information, a well-studied measure in information theory introduced by Wyner (IEEE Trans. Inf. 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 etal. (SIAM J. Comput., 32 (6):1570-1585, 2003).