A Closed-Form Shave from Occam's Quantum Razor: Exact Results for Quantum Compression

TL;DR

This paper derives a closed-form expression for quantum compression efficiency of stochastic processes, revealing how quantum advantage depends on process cryptic order and enabling precise calculations across all code lengths.

Contribution

It introduces an exact spectral decomposition method to compute quantum communication costs for any code length, including infinite, advancing understanding of quantum compression of complex processes.

Findings

Quantum advantage increases with codeword length.

Finite-codeword compression is governed by the process's cryptic order.

The method allows exact calculation of quantum costs for infinite processes.

Abstract

The causal structure of a stochastic process can be more efficiently transmitted via a quantum channel than a classical one, an advantage that increases with codeword length. While previously difficult to compute, we express the quantum advantage in closed form using spectral decomposition, leading to direct computation of the quantum communication cost at all encoding lengths, including infinite. This makes clear how finite-codeword compression is controlled by the classical process' cryptic order and allows us to analyze structure within the length-asymptotic regime of infinite-cryptic order (and infinite Markov order) processes.