Dynamics of initially entangled open quantum systems (14 pages)

Claudio Altafini

Jonas Glatthard

The entanglement entropy of a black hole and that of its Hawking radiation are expected to follow the so-called Page curve: After an increase in line with Hawking’s calculation, it is expected to decrease back to zero once the black hole has fully evaporated, as demanded by unitarity. Recently, a simple system-plus-bath model has been proposed which shows a similar behavior. Here, we make a general argument as to why such a Page-curve-like entanglement dynamics should be expected to hold generally for system-plus-bath models at small coupling and low temperatures, when the system is initialized in a pure state far from equilibrium. The interaction with the bath will then generate entanglement entropy, but it eventually has to decrease to the value prescribed by the corresponding mean-force Gibbs state. Under those conditions, it is close to the system ground state. We illustrate this on two paradigmatic open-quantum-system models, the exactly solvable harmonic quantum Brownian motion and the spin-boson model, which we study numerically. In the first example we find that the intermediate entropy of an initially localized impurity is higher for more localized initial states. In the second example, for an impurity initialized in the excited state, the Page time—when the entropy reaches its maximum—occurs when the excitation has half decayed. Published by the American Physical Society 2024

V. M. Akulin

Andreas Osterloh, Luigi Amico A, Francesco Plastina

Jonas Glatthard

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.

Mike Blake, Anthony P. Thompson

Abstract We study entanglement dynamics in toy models of black hole information built out of chaotic many-body quantum systems, by utilising a coarse-grained description of entanglement dynamics in such systems known as the ‘entanglement membrane’. We show that in these models the Page curve associated to the entropy of Hawking radiation arises from a transition in the entanglement membrane around the Page time, in an analogous manner to the change in quantum extremal surfaces that leads to the Page curve in semi-classical gravity. We also use the entanglement membrane prescription to study the Hayden-Preskill protocol, and demonstrate how information initially encoded in the black hole is rapidly transferred to the radiation around the Page time. Our results relate recent developments in black hole information to generic features of entanglement dynamics in chaotic many-body quantum systems.