Correlation/Communication complexity of generating bipartite states

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We study the correlation complexity (or equivalently, the communication complexity) of generating a bipartite quantum state $\rho$.

• When $\rho$ is a pure state, we completely characterize the complexity for approximately generating $\rho$ 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 $\rho$ 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 $\rho$.