A universal coding scheme for remote generation of continuous random variables

Abstract

We consider a setup in which Alice selects a pdf ƒ from a set of prescribed pdfs P and sends a prefix-free codeword W to Bob in order to allow him to generate a single instance of the random variable X ∼ ƒ. We describe a universal coding scheme for this setup and establish an upper bound on the expected codeword length when the pdf ƒ is bounded, orthogonally concave (which includes quasiconcave pdf), and has a finite first absolute moment. A dyadic decomposition scheme is used to express the pdf as a mixture of uniform pdfs over hypercubes. Alice randomly selects a hypercube according to its weight, encodes its position and size into W, and sends it to Bob who generates X uniformly over the hypercube. Compared to previous results on channel simulation, our coding scheme applies to any continuous distribution and does not require two-way communication or shared randomness. We apply our coding scheme to classical simulation of quantum entanglement and obtain a better bound on the average codeword length than previously known.