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arXiv:1203.1153

Correlation/Communication complexity of generating bipartite states

Rahul Jain, Yaoyun Shi, Zhaohui Wei, Shengyu Zhang

Published Mar 6, 2012• 5 citations
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Abstract

We study the correlation complexity (or equivalently, the communication complexity) of generating a bipartite quantum state $ρ$. When $ρ$ is a pure state, we completely characterize the complexity for approximately generating $ρ$ by a corresponding approximate rank, closing a gap left in Ambainis, Schulman, Ta-Shma, Vazirani and Wigderson (SIAM Journal on Computing, 32(6):1570-1585, 2003). When $ρ$ is a classical distribution $P(x,y)$, we tightly characterize the complexity of generating $P$ by the psd-rank, a measure recently proposed by Fiorini, Massar, Pokutta, Tiwary and de Wolf (STOC 2012). We also present a characterization of the complexity of generating a general quantum state $ρ$.

DOI
10.48550/arxiv.1203.1153

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We study the correlation complexity (or equivalently, the communication complexity) of generating a bipartite quantum state ρρρ. When ρρρ is a pure state, we completely characterize the complexity for approximately generating ρρρ by a corresponding approximate rank, closing a gap left in Ambainis, Schulman, Ta-Shma, Vazirani and Wigderson (SIAM Journal on Computing, 32(6):1570-1585, 2003). When ρρρ is a classical distribution P(x,y)P(x,y)P(x,y), we tightly characterize the complexity of generating PPP by the psd-rank, a measure recently proposed by Fiorini, Massar, Pokutta, Tiwary and de Wolf (STOC 2012). We also present a characterization of the complexity of generating a general quantum state ρρρ.
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We consider a realistic setting of quantum tasks that generate shared bipartite quantum states. Suppose \alice and \bob are located at different places and need to produce a target shared quantum state ρρρ. In order to save quantum communication, they can choose to share a proper smaller quantum state σσσ first, and then turn σσσ to ρρρ by performing only local quantum operations and classical communications (LOCC). We hope σσσ is the optimal such that the quantum communication needed is as little as possible, which is called the generation cost of ρρρ. In this paper, for an arbitrary bipartite ρρρ, we characterize its generation cost completely by proving that it is exactly equivalent to the logarithm of the Schmidt number of ρρρ. Similar quantum schemes where classical communication is not allowed have actually been considered. By comparing the two settings, we are able to look into the role that classical communication plays in these fundamental tasks, where we exhibit some instances in which classical communication is not helpful completely.
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