The classical-quantum divergence of complexity in modelling spin chains

TL;DR

This paper investigates how quantum mechanics can alter the behavior of the statistical complexity in modeling spin chains, showing that quantum models can exhibit non-monotonic complexity growth with temperature, unlike classical models.

Contribution

The study constructs the simplest quantum model for the Ising spin chain and reveals a fundamentally different, non-monotonic complexity behavior compared to classical models.

Findings

Quantum statistical complexity peaks at finite temperature.

Quantum models show complexity decreases at high temperatures.

Classical complexity grows monotonically with temperature.

Abstract

The minimal memory required to model a given stochastic process - known as the statistical complexity - is a widely adopted quantifier of structure in complexity science. Here, we ask if quantum mechanics can fundamentally change the qualitative behaviour of this measure. We study this question in the context of the classical Ising spin chain. In this system, the statistical complexity is known to grow monotonically with temperature. We evaluate the spin chain's quantum mechanical statistical complexity by explicitly constructing its provably simplest quantum model, and demonstrate that this measure exhibits drastically different behaviour: it rises to a maximum at some finite temperature then tends back towards zero for higher temperatures. This demonstrates how complexity, as captured by the amount of memory required to model a process, can exhibit radically different behaviour when quantum processing is allowed.