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arXiv:2401.06042 · Physical review. D/Physical review. D.

Page-curve-like entanglement dynamics in open quantum systems

Jonas Glatthard

Published Apr 11, 2024• 9 citations
arXivHTMLPDF

Abstract

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

DOI
10.1103/physrevd.109.l081901

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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
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.
The entanglement entropy of black holes is expected to follow the Page curve. After an initial linear increase with time the entanglement entropy should reach a maximum at the Page time and then decrease. This paper introduces an exactly solvable model of free fermions that explicitly shows such a Page curve: The entanglement entropy vanishes asymptotically for late times instead of saturating at a volume law. The bending down of the Page curve is accompanied by a breakdown of the semiclassical connection between particle current and entanglement generation, a quantum phase transition in the entanglement Hamiltonian and nonanalytic behavior of the $q\ensuremath{\rightarrow}\ensuremath{\infty}$ R\'enyi entropy. These observations are expected to hold for a larger class of systems beyond the exactly solvable model analyzed here.
A bstract If the evaporation of a black hole formed from a pure state is unitary, the entanglement entropy of the Hawking radiation should follow the Page curve, increasing from zero until near the halfway point of the evaporation, and then decreasing back to zero. The general argument for the Page curve is based on the assumption that the quantum state of the black hole plus radiation during the evaporation process is typical. In this paper, we show that the Page curve can result from a simple dynamical input in the evolution of the black hole, based on a recently proposed signature of quantum chaos, without resorting to typicality. Our argument is based on what we refer to as the “operator gas” approach, which allows one to understand the evolution of the microstate of the black hole from generic features of the Heisenberg evolution of operators. One key feature which leads to the Page curve is the possibility of dynamical processes where operators in the “gas” can “jump” outside the black hole, which we refer to as void formation processes. Such processes are initially exponentially suppressed, but dominate after a certain time scale, which can be used as a dynamical definition of the Page time. In the Hayden-Preskill protocol for young and old black holes, we show that void formation is also responsible for the transfer of information from the black hole to the radiation. We conjecture that void formation may provide a microscopic explanation for the recent semi-classical prescription of including islands in the calculation of the entanglement entropy of the radiation.
We study the dynamics of entanglement in a one-dimensional XXZXXZXXZ spin-1/21/21/2 chain with and without integrability-breaking interactions and connected to a bath. We start from a state where the system and bath are completely unentangled, and the bath is polarized spin-down. We consider two different initial states for the system: (i) a polarized spin-up state, and (ii) an infinite-temperature state. In the particle representation of the spin chain, the polarized spin-up state corresponds to a filled state, while the polarized spin-down state corresponds to an empty state. Starting from these inhomogeneous quenches, in all the above-mentioned cases, we obtain the Page curve like behavior in the entanglement. We report different power-law behavior in the growth of entanglement for different initial states and different kinds of baths (interacting and noninteracting). In an attempt to explore plausible deep connections between entanglement and Boltzmann entropy, we investigate the latter in both the filled and the infinite-temperature case, for the system and the bath. For the filled case, the Boltzmann entropy of the system has the form of a Page curve but quantitatively deviates from the entanglement. On the other hand, the entropy of the bath keeps increasing. Remarkably, for the infinite-temperature case, we find that the system and bath Boltzmann entropies agree with the entanglement entropy, after and before the Page time, respectively. Our findings are expected to hold for generic interacting quantum systems and could be of relevance to black-hole physics.
We investigate the multi-partite entanglement structure of an evaporating black hole and its Hawking radiation by dividing the radiation into finer subsystems. We approximate an evaporating black hole and its radiation with a Haar-random state for this purpose. Using the multi-entropy of these configurations, we define a black hole multi-entropy curve, which describes how the multi-entropy changes during the black hole evaporation. This black hole multi-entropy curve is a natural generalization of the Page curve since the multi-entropy reduces to the entanglement entropy for the bi-partite case. The multi-entropy curve keeps increasing in the early time. It reaches the maximum value at the multi-entropy time, which is later than the Page time, and starts to decrease. However, it does not decrease to zero at the end of the black hole evaporation. This non-zero value of the multi-entropy represents the secret entanglement between Hawking particles.
A bstract We investigate the multi-partite entanglement structure of an evaporating black hole and its Hawking radiation by dividing the radiation into finer subsystems. We approximate an evaporating black hole and its radiation with a Haar-random state for this purpose. Using the multi-entropy of these configurations, we define a black hole multi-entropy curve , which describes how the multi-entropy changes during the black hole evaporation. This black hole multi-entropy curve is a natural generalization of the Page curve since the multi-entropy reduces to the entanglement entropy for the bi-partite case. The multi-entropy curve keeps increasing in the early time. It reaches the maximum value at the multi-entropy time , which is later than the Page time, and starts to decrease. However, it does not decrease to zero at the end of the black hole evaporation. This non-zero value of the multi-entropy represents the secret entanglement between Hawking particles.