Thermodynamics of the Page curve in Markovian open quantum systems

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.