Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states

We prove that the entanglement entropy of any pure initial state of a bipartite bosonic quantum system grows linearly in time with respect to the dynamics induced by any unstable quadratic Hamiltonian. The growth rate does not depend on the initial state and is equal to the sum of certain Lyapunov exponents of the corresponding classical dynamics. This paper generalizes the findings of [Bianchi et al., JHEP 2018, 25 (2018)], which proves the same result in the special case of Gaussian initial states. Our proof is based on a recent generalization of the strong subadditivity of the von Neumann entropy for bosonic quantum systems [De Palma et al., arXiv:2105.05627]. This technique allows us to extend our result to generic mixed initial states, with the squashed entanglement providing the right generalization of the entanglement entropy. We discuss several applications of our results to physical systems with (weakly) interacting Hamiltonians and periodically driven quantum systems, including certain quantum field theory models.

Unitarity and Page Curve for Evaporation of 2D AdS Black Holes

We explore the Hawking evaporation of two-dimensional anti-de Sitter (AdS2), dilatonic black hole coupled with conformal matter, and derive the Page curve for the entanglement entropy of radiation. We first work in a semiclassical approximation with backreaction. We show that the end-point of the evaporation process is AdS2 with a vanishing dilaton, i.e., a regular, singularity-free, zero-entropy state. We explicitly compute the entanglement entropies of the black hole and the radiation as functions of the horizon radius, using the conformal field theory (CFT) dual to AdS2 gravity. We use a simplified toy model, in which evaporation is described by the forming and growing of a negative mass configuration in the positive-mass black hole interior. This is similar to the “islands” proposal, recently put forward to explain the Page curve for evaporating black holes. The resulting Page curve for AdS2 black holes is in agreement with unitary evolution. The entanglement entropy of the radiation initially grows, closely following a thermal behavior, reaches a maximum at half-way of the evaporation process, and then goes down to zero, following the Bekenstein–Hawking entropy of the black hole. Consistency of our simplified model requires a non-trivial identification of the central charge of the CFT describing AdS2 gravity with the number of species of fields describing Hawking radiation.

Failure of the split property in gravity and the information paradox

In an ordinary quantum field theory, the “split property” implies that the state of the system can be specified independently on a bounded subregion of a Cauchy slice and its complement. This property does not hold for theories of gravity, where observables near the boundary of the Cauchy slice uniquely fix the state on the entire slice. The original formulation of the information paradox explicitly assumed the split property and we follow this assumption to isolate the precise error in Hawking’s argument. A similar assumption also underpins the monogamy paradox of Mathur and AMPS. Finally the same assumption is used to support the common idea that the entanglement entropy of the region outside a black hole should follow a Page curve. It is for this reason that computations of the Page curve have been performed only in nonstandard theories of gravity, which include a nongravitational bath and massive gravitons. The fine-grained entropy at I^{+} does not obey a Page curve for an evaporating black hole in standard theories of gravity but we discuss possibilities for coarse graining that might lead to a Page curve in such cases.

Defect extremal surface for reflected entropy

Defect extremal surface is defined by extremizing the Ryu-Takayanagi formula corrected by the quantum defect theory. This is interesting when the AdS bulk contains a defect brane (or string). We introduce a defect extremal surface formula for reflected entropy, which is a mixed state generalization of entanglement entropy measure. Based on a decomposition procedure of an AdS bulk with a brane, we demonstrate the equivalence between defect extremal surface formula and island formula for reflected entropy in AdS3/BCFT2. We also compute the evolution of reflected entropy in evaporating black hole model and find that defect extremal surface formula agrees with island formula.