Correlation/Communication complexity of generating bipartite states

TL;DR

This paper characterizes the correlation and communication complexity of generating bipartite quantum states, providing exact measures for pure and classical states and a general framework for quantum states.

Contribution

It offers a complete characterization of the complexity for pure states via approximate rank, and for classical distributions via psd-rank, extending to general quantum states.

Findings

Complexity for pure states characterized by approximate rank.

Complexity for classical distributions characterized by psd-rank.

General quantum state complexity characterized in the paper.

Abstract

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 $\rho$.