Focus issue: Entanglement and quantum gravity

We review recent results regarding entanglement in quantum fields in cosmological spacetimes and related phenomena in flat spacetime such as the Unruh effect. We begin with a summary of important results about field entanglement and the mathematics of Bogoliubov transformations that is very often used to describe it. We then discuss the Unruh–DeWitt detector model, which is a useful model of a generic local particle detector. This detector model has been successfully used as a tool to obtain many important results. In this context we discuss two specific types of these detectors: a qubit and a harmonic oscillator. The latter has recently been shown to have important applications when one wants to probe nonperturbative physics of detectors interacting with quantum fields. We then detail several recent advances in the study and application of these ideas, including echoes of the early universe, entanglement harvesting, and a nascent proposal for quantum seismology.

We propose entanglement entropy as a probe of the architecture of spacetime in quantum gravity. We argue that the leading contribution to this entropy satisfies an area law for any sufficiently large region in a smooth spacetime, which, in fact, is given by the Bekenstein–Hawking formula. This conjecture is supported by various lines of evidence from perturbative quantum gravity, simplified models of induced gravity, the AdS/CFT correspondence and loop quantum gravity, as well as Jacobsonʼs 'thermodynamic' perspective of gravity.

Entanglement entropy has proven to be an extremely useful concept in quantum field theory. Gauge theories are of particular interest, but for these systems the entanglement entropy is not clearly defined because the physical Hilbert space does not factor as a tensor product according to regions of space. Here we review a definition of entanglement entropy that applies to abelian and nonabelian lattice gauge theories. This entanglement entropy is obtained by embedding the physical Hilbert space into a product of Hilbert spaces associated to regions with boundary. The latter Hilbert spaces include degrees of freedom on the entangling surface that transform like surface charges under the gauge symmetry. These degrees of freedom are shown to contribute to the entanglement entropy, and the form of this contribution is determined by the gauge symmetry. We test our definition using the example of two-dimensional Yang–Mills theory, and find that it agrees with the thermal entropy in de Sitter space, and with the results of the Euclidean replica trick. We discuss the possible implications of this result for more complicated gauge theories, including quantum gravity.

In an approach to quantum gravity where space-time arises from coarse graining of fundamentally discrete structures, black hole formation and subsequent evaporation can be described by a unitary evolution without the problems encountered by the standard remnant scenario or the schemes where information is assumed to come out with the radiation during evaporation (firewalls and complementarity). The final state is purified by correlations with the fundamental pre-geometric structures (in the sense of Wheeler), which are available in such approaches, and, like defects in the underlying space-time weave, can carry zero energy.

It has been argued that the entropy computed in the isolated horizon framework of loop quantum gravity is closely related to the entanglement entropy of the gravitational field, and that the calculation performed is not restricted to horizons. We recall existing work on this issue and explain how recent work on generalizing these computations to arbitrary spacetime dimensions $D+1\geqslant 3$ supports this point of view and makes the duality between entanglement entropy and the entropy computed from counting boundary states manifest. In a certain semiclassical regime in $3+1$ dimensions, this entropy is given by the Bekenstein–Hawking formula.

We discuss the possible relevance of complex codimension-two extremal surfaces to the Ryu–Takayanagi holographic entanglement proposal and its covariant Hubeny–Rangamani–Takayanagi generalization. Such surfaces live in a complexified bulk spacetime defined by analytic continuation. We identify surfaces of this type for BTZ, Schwarzschild–AdS, and Schwarzschild–Lifshitz planar black holes. Since the dual CFT interpretation for the imaginary part of their areas is unclear, we focus on a straw man proposal relating CFT entropy to the real part of the area alone. For Schwarzschild–AdS and Schwarzschild–Lifshitz, we identify families where the real part of the area agrees with qualitative physical expectations for the time-dependence of the appropriate CFT entropy and, in addition, where it is smaller than the area of corresponding real extremal surfaces. It is thus plausible that the CFT entropy is controlled by these complex extremal surfaces.

Sorkin (Expressing entropy globally in terms of (4D) field-correlations arXiv:1205.2953) defines an entropy for a Gaussian scalar field ϕ in an arbitrary region of either a continuous spacetime or a causal set, given only the correlator $\langle \phi (x)\phi (y)\rangle$ within the region. The definition is global and independent of any choice of spacelike hypersurface. As a first application, we compute numerically the entanglement entropy in two cases where the asymptotic form is known or suspected from conformal field theory, finding excellent agreement when the required ultraviolet cutoff is implemented as a truncation on spacetime mode sums. We also show how the symmetry of entanglement entropy reflects the fact that RS and SR share the same eigenvalues, with R and S being arbitrary matrices.

Maldacena and Susskind have proposed a correspondence between wormholes and entanglement, dubbed ER=EPR. We study this in the context of three-dimensional topological quantum field theory (TQFT), where we show that the formation of a wormhole is the same process as creating a particle–antiparticle pair. A key feature of the ER=EPR proposal is that certain apparently entangled degrees of freedom turn out to be the same. We name this phenomenon 'fake entanglement', and show how it arises in our TQFT model.