A Universal Coding Scheme for Remote Generation of Continuous Random Variables

Alice selects an arbitrary pdf f and uses a stochastic encoder to generate a prefix-free codeword M, which is sent to Bob so that he can generate a single instance of the random variable X ~ f. We describe a universal coding scheme for this setup which works for any f , and establish an upper bound on its expected codeword length when the pdf f 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 M, 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. Applying our coding scheme to classical simulation of quantum entanglement, we obtain a tighter bound on the average codeword length than previously known.