Stability and area law for rapidly mixing quantum dissipative systems

TL;DR

This thesis demonstrates that rapidly mixing open quantum systems are stable against perturbations and their fixed points obey an area law for mutual information, revealing key properties of dissipative quantum dynamics.

Contribution

It establishes that rapid mixing implies stability and an area law for the fixed point in dissipative quantum spin systems.

Findings

Rapid mixing ensures stability against external perturbations.

Fixed points of rapidly mixing systems satisfy an area law for mutual information.

The mixing time scales logarithmically with system size.

Abstract

In this thesis we study properties of open quantum dissipative evolutions of spin systems on lattices described by Lindblad generators, in a particular regime that we denote rapid mixing. We consider dissipative evolutions with a unique fixed point, and which compress the whole space of input states into increasingly small neighborhoods of the fixed point. The time scale at which this compression takes place, or in other words the time we have to wait for any input state to become almost indistinguishable from the fixed point, is called the mixing time of the process. Rapid mixing is a condition on the scaling of this mixing time with the system size: if it is logarithmic, then we have rapid mixing. The main contribution of this thesis is to show that rapid mixing has profound implications for the corresponding system: it is stable against external perturbations and its fixed point satisfies an area law for mutual information.

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