A Universal Coding Scheme for Remote Generation of Continuous Random Variables

摘要

We consider a setup in which Alice selects a pdf $f$ from a set of prescribedpdfs $\mathscr{P}$ and sends a prefix-free codeword $W$ to Bob in order toallow him to generate a single instance of the random variable $X\sim f$. Wedescribe a universal coding scheme for this setup and establish an upper boundon the expected codeword length when the pdf $f$ is bounded, orthogonallyconcave (which includes quasiconcave pdf), and has a finite first absolutemoment. A dyadic decomposition scheme is used to express the pdf as a mixtureof uniform pdfs over hypercubes. Alice randomly selects a hypercube accordingto its weight, encodes its position and size into $W$, and sends it to Bob whogenerates $X$ uniformly over the hypercube. Compared to previous results onchannel simulation, our coding scheme applies to any continuous distributionand does not require two-way communication or shared randomness. We apply ourcoding scheme to classical simulation of quantum entanglement and obtain abetter bound on the average codeword length than previously known.