A Universal Coding Scheme for Remote Generation of Continuous Random Variables

TL;DR

This paper introduces a universal coding scheme enabling Alice to efficiently communicate a continuous random variable to Bob, applicable to any bounded, orthogonally concave pdf, with improved bounds in quantum entanglement simulation.

Contribution

The paper presents a novel universal coding scheme for continuous distributions that does not require shared randomness or two-way communication, improving bounds in quantum entanglement simulation.

Findings

Upper bound on expected codeword length for bounded, orthogonally concave pdfs

Applicable to any continuous distribution without shared randomness

Improved bounds in quantum entanglement simulation

Abstract

We consider a setup in which Alice selects a pdf $f$ from a set of prescribed pdfs $\mathscr{P}$ and sends a prefix-free codeword $W$ to Bob in order to allow him to generate a single instance of the random variable $X\sim f$. We describe a universal coding scheme for this setup and establish an upper bound on the 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 $W$, 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.