Thermalization with partial information

TL;DR

This paper introduces a maximum channel entropy principle to model quantum system dynamics, extending thermodynamic concepts to noisy quantum channels and proposing a learning algorithm for such channels.

Contribution

It formulates a new principle for identifying quantum channels based on maximum entropy, generalizes the microcanonical derivation to channels, and develops a learning algorithm for these channels.

Findings

Derived the mathematical structure of the maximum entropy quantum channel

Established a postselection theorem relating permutation-invariant channels to i.i.d. channels

Proposed a learning algorithm for quantum channels based on the maximum entropy principle

Abstract

A many-body system, whether in contact with a large environment or evolving under complex dynamics, can typically be modeled as occupying the thermal state singled out by Jaynes' maximum entropy principle. Here, we find analogous fundamental principles identifying a noisy quantum channel $\mathcal{T}$ to model the system's dynamics, going beyond the study of its final equilibrium state. Our maximum channel entropy principle states that $\mathcal{T}$ should maximize the channel's entropy, suitably defined, subject to any available macroscopic constraints. These may correlate input and outputs, and may lead to restricted or partial thermalizing dynamics including thermalization with average energy conservation. This principle is reinforced by an independent extension of the microcanonical derivation of the thermal state to channels, which leads to the same $\mathcal{T}$. Our technical contributions include a derivation of the general mathematical structure of $\mathcal{T}$, a custom postselection theorem relating an arbitrary permutation-invariant channel to nearby i.i.d. channels, as well as novel typicality results for quantum channels for noncommuting constraints and arbitrary input states. We propose a learning algorithm for quantum channels based on the maximum channel entropy principle, demonstrating the broader relevance of $\mathcal{T}$ beyond thermodynamics and complex many-body systems.