Thermodynamics of the Page curve in Markovian open quantum systems

TL;DR

This paper investigates the entropy dynamics of open quantum systems described by Lindbladian evolution, revealing conditions under which the subsystem entropy exhibits Page curve behavior and linking entropy decrease to heat flow, with analytic results for specific models.

Contribution

It provides an analytic description of the Page curve in Markovian open quantum systems and connects entropy dynamics to thermodynamic principles.

Findings

Page time occurs when half the initial energy leaves the system.

Entropy can temporarily exceed the final thermal value in certain regimes.

Analytic solutions for entanglement dynamics in two-level systems and oscillators.

Abstract

Typically, the von Neumann entropy of a subsystem increases until it plateaus at the thermal value. Under some circumstances, however, the intermediate value can dwarf the final value, even if the subsystem starts in a pure state. A famous example in the context of the black hole information paradox is the entropy of the Hawking radiation, where this behaviour is dubbed the Page curve. More generally, this is the case for excited systems weakly coupled to cold reservoirs. Here we study the entropy dynamics for Lindbladian evolution, i.e. open quantum systems in weak contact with Markovian reservoirs. This allows us to study the non-equilibrium thermodynamics of the subsystem entropy decrease and link it to Landauer's principle: the entropy decrease must be accompanied by a heat flow out of the system. We give an analytic expression of the entanglement dynamics for a decaying excitation in a two-level system and study it under equilibration of a localised oscillator. In both cases the Page time occurs when half the initial energy has left the system.