DocumentCode :
62722
Title :
Efficient Protocols for Generating Bipartite Classical Distributions and Quantum States
Author :
Jain, R. ; Yaoyun Shi ; Zhaohui Wei ; Shengyu Zhang
Author_Institution :
Dept. of Comput. Sci., Nat. Univ. of Singapore, Singapore, Singapore
Volume :
59
Issue :
8
fYear :
2013
fDate :
Aug. 2013
Firstpage :
5171
Lastpage :
5178
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 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).
Keywords :
optimisation; protocols; quantum communication; TSP; bipartite classical distributions; bipartite quantum state generation; communication cost; convex geometry; efficient communication protocols; exponential gap; information theory; nonnegative rank; optimal protocol; optimization problems; quantum communication; quantum states; Approximation methods; Complexity theory; Correlation; Information theory; Protocols; Quantum mechanics; Random variables; Communication complexity; probability distribution generation; psd-rank; quantum correlation complexity;
fLanguage :
English
Journal_Title :
Information Theory, IEEE Transactions on
Publisher :
ieee
ISSN :
0018-9448
Type :
jour
DOI :
10.1109/TIT.2013.2258372
Filename :
6516589
Link To Document :
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