Advances in High Energy Physics

Volume 2017, Article ID 8621513, 7 pages

https://doi.org/10.1155/2017/8621513

## Entropic Entanglement: Information Prison Break

Space Research and Technology Institute, Bulgarian Academy of Sciences, Sofia, Bulgaria

Correspondence should be addressed to Alexander Y. Yosifov; moc.liamg@vofisoyyrednaxela

Received 13 January 2017; Accepted 19 April 2017; Published 4 July 2017

Academic Editor: George Siopsis

Copyright © 2017 Alexander Y. Yosifov and Lachezar G. Filipov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We argue that certain nonviolent local quantum field theory (LQFT) modification considered at the global horizon of a static spherically symmetric black hole can lead to adiabatic leakage of quantum information in the form of Hawking particles. The source of the modification is (i) smooth at and (ii) rapidly vanishing at . Furthermore, we restore the unitary evolution by introducing extra quanta which departs slightly from the generic Hawking emission without changing the experience of an infalling observer (no drama). Also, we suggest that a possible interpretation of the Bekenstein-Hawking bound as entanglement entropy may yield a nonsingular dynamical horizon behavior described by black hole thermodynamics. Hence, by treating gravity as a field theory and considering its coupling to the matter fields in the Minkowski vacuum, we derive the conjectured fluctuations of the background geometry of a black hole.

#### 1. Introduction

It has been argued in [1] that black holes are not black at all. Rather, in a semiclassical approximation they are shown to be hot bodies that emit thermal radiation with an inverse temperature of . The fate of information fallen into a black hole is still under debate. Hawking’s original proposal of loss of information and thus pure-to-mixed state evolution has been strongly opposed [2–11] as it implies violation of quantum-mechanical unitarity. Resolution of the information paradox within the current nomenclature does not seem to be a fruitful endeavor. Instead, we have focused on modifying already existing principles. Abandoning locality above the Planck mass (), for instance, appears as a promising and somewhat more conservative approach. Significant theoretical support for fundamental nonlocality has come from AdS/CFT duality and cosmology [12–16]. One of the authors [16] has shown that in the extreme conditions of the early universe (super-Planckian energies) nonlocality plays an essential role for explaining the origin of the cosmological principle.

Following the theoretical evidence, in the current paper, we embrace the notion of locality as an effective field theory, manifesting in weak gravitational dynamics. Based on that assumption we propose a framework, featuring a modification of local quantum field theory as defined on the global horizon in order to provide a nonviolent mechanism for taking the Hawking quanta (quantum information) out of the hole, and thus restore unitarity. The suggested LQFT modification leads to weak (nonviolent) quantum effects which manifest in brief nonlocal phenomena. Despite being weak, the perturbative nature of the effects makes them significant in the course of a black hole’s lifetime . The current model also predicts minor deviations from the generic Hawking emission sufficient to restore the unitary evolution without causing drama for an observer in free fall.

By treating gravity in a black hole background metric (Minkowski space) as a field theory* (graviton)* we derive the microscopic origin of the conjectured Planckian-amplitude horizon oscillations [17]. In a previous paper we approached the phenomena classically by deriving the oscillations from perturbation theory (see [17]). The results we obtain in the particular work may be considered as a microscopic origin of the stretched horizon in observer complementarity [18]. Furthermore, the effects of the source of the LQFT modification are shown to respect the equivalence principle and also to rapidly vanish at large* r*.

The paper is organized as follows. In Section 2 we put forward the LQFT revisions and show how the coupling between the graviton and the matter fields in Minkowski space can lead to information escape. In Section 3 we provide a quantum theory derivation of the proposed horizon oscillations by treating gravity in the vicinity of a black hole as field theory.

#### 2. Nonlocal Information Release

The particular modification of LQFT, which we propose, comes from localized and brief violations of locality, yielded by “strong” fluctuations of the* graviton* that come from its coupling to the matter fields in Minkowski space. We also put forward a gedanken experiment which involves a pair of strings put on both sides of the future Rindler horizon in order to show that the equivalence principle remains valid for an observer in free fall.

Based on the thermal spectrum of the emitted radiation and black hole thermodynamics (Second Law, in particular) we now take for granted the proportionality between black hole entropy and horizon area. Namely, the entropy of a black hole is one-fourth of the area of the event horizon in Planck units:

The geometric entropy bound is deeply rooted in holography and further generalized in the Bousso bound [19]. Yet, its origin has not been fully explained.

##### 2.1. Geometric Entropy = Entanglement Entropy

We begin by showing how the Bekenstein formula can be derived from entanglement entropy and later how this can yield the small departures from LQFT needed to carry the quantum information out of the black hole.

Bianchi has shown [20] that by considering the correlations between gravity and matter fields in the near-horizon region one can reproduce the Bekenstein-Hawking bound equation (1). The obtained equality has been shown to be universal and independent of the number of field* species*. Hence Bianchi’s derivation of the equality , where is entanglement entropy, shows that quantum entanglement is the fundamental origin of the Bekenstein entropy bound. Let us further clarify that.

Imagine we have a Schwarzschild black hole in a pure state with metric where the singularity is at and the global horizon is at .

Here, a black hole event horizon provides a perfect entangling surface as it naturally causally disconnects the interior and exterior regions where is the interior region and is the exterior . The dimensionality of is given as the logarithm of the internal degrees of freedom.

The pure state of the complete system is given by the product of the two subsystems with a corresponding density matrix

The pure state of the complete system may be decomposed as where and denote the reduced density matrices of the corresponding subsystems and , respectively. Note that in a black hole background the initial state cannot be trivially reproduced by the thermal density matrices; some of the information concerning is found in the entanglement between the two subsystems across the entangling surface.

Consider the Minkowski vacuum in the region near the black hole which is bounded by a local Rindler horizon * *. The complementary* left* and* right* Rindler wedges, described by the Hilbert spaces and , respectively, are given in terms of thermal density matrices; see (6). The complete vacuum state is due to the entangling between the field theories defined on both sides of the horizon (*L* and Rindler wedges): where and are the eigenstates associated with the wedges and denotes the inverse temperature.

In the vacuum state every mode on the* left* Rindler wedge is entangled with the corresponding mode on the* right* wedge. The particular entanglement normalizes the stress-energy tensor at , and hence an infalling observer does not feel anything out of the ordinary. The stress tensor normalization provides a smooth transition between the two distinct causal patches. Thus the entanglement entropy is proportional to the entangling surface (horizon). As a result, only modes very close to the global horizon contribute to the entropy of the system.

Bianchi’s derivation of the equality strongly advocates the entanglement origin of the geometric entropy. Considering the correlations between gravity and matter fields in Minkowski space suggests we can treat gravity as a field theory* (graviton)*. In particular, we argue that by treating gravity in Minkowski space in terms of quantum field theory in black hole background we can obtain the desired modification of local quantum field theory in the vicinity of the horizon and thus present a framework for gradual release of quantum information.

##### 2.2. Gravity as a Field Theory

Suppose we assign a time-dependent Killing frequency to the* graviton* with respect to the background metric. The correlated quantum fields (gravity and matter) in the near-horizon region oscillate rapidly and have radial dependence with respect to the Rindler horizon * *. Generally, the quantum fluctuations of the matter fields near the horizon get “amplified” by the black hole’s internal degrees of freedom and an inertial observer with a measuring apparatus in that region measures , where the expectation value of is nonzero (Hawking process).

We wish to focus on the black hole metric back-reaction from the* graviton* fluctuations in two cases (i) Minkowski space and (ii) the vicinity of the horizon. Since we consider locality in effective field theory to be a constraint imposed by the background geometry, we wish to examine how fluctuations of the gravitational field above a threshold affect it.

We believe that strong quantum fluctuations of the gravitational field as considered* onto* the horizon will cause “disturbances” in the background metric and the effective field theory description (locality). Consider the following gedanken experiment. Imagine we have a static black hole; see (2) and Figure 1.