Decoherence Time Scales and the H\"ormander condition

TL;DR

This paper investigates how decoherence propagates in open quantum systems using the H"ormander condition, revealing delays in decoherence onset and methods to identify decoherence-free subsystems.

Contribution

It introduces the use of the H"ormander condition to analyze decoherence timescales and the spread of noise in quantum systems, linking PDE hypoellipticity to quantum decoherence.

Findings

H"ormander condition determines decoherence delay in Gaussian channels

The condition helps identify decoherence-free subsystems

Decoherence propagates through system degrees of freedom based on commutator structure

Abstract

We consider an open quantum system described by the GKLS equation and we are interested in the onset of decoherence. We are in particulary interested in situations where only some degrees of freedom of the system are coupled to the environment, and we want to understand if, and how fast, the noise travels through the system and eventually affects all degrees of freedom. We find that this can be understood in terms of the H\"ormander condition, a condition on the commutators of the Hamiltonian vectorfields of the Lindblad operators and the internal Hamiltonian, which is a condition for hypoellipticity known from the theory of PDE's. We show that for Gaussian quantum channels this condition leads to a delay in the onset of decoherence and can as well be used to detect decoherence free subsystems.