Occam's Quantum Strop: Synchronizing and Compressing Classical Cryptic Processes via a Quantum Channel

TL;DR

This paper explores how quantum channels can significantly reduce the information needed to synchronize and compress classical stochastic processes, leveraging quantum state-indistinguishability and extended causal structures.

Contribution

It generalizes quantum advantage in process synchronization, introduces an algorithm to compute this advantage, and relates maximum compression to the process's cryptic order.

Findings

Quantum advantage increases with extended causal structures

Maximum compression is linked to the process's cryptic order

An efficient algorithm computes the quantum advantage

Abstract

A stochastic process's statistical complexity stands out as a fundamental property: the minimum information required to synchronize one process generator to another. How much information is required, though, when synchronizing over a quantum channel? Recent work demonstrated that representing causal similarity as quantum state-indistinguishability provides a quantum advantage. We generalize this to synchronization and offer a sequence of constructions that exploit extended causal structures, finding substantial increase of the quantum advantage. We demonstrate that maximum compression is determined by the process's cryptic order---a classical, topological property closely allied to Markov order, itself a measure of historical dependence. We introduce an efficient algorithm that computes the quantum advantage and close noting that the advantage comes at a cost---one trades off prediction for generation complexity.