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arXiv:1311.3598 · International Journal of Quantum Information

A statistical model of information evaporation of perfectly reflecting black holes

László Gyöngyösi

Published Nov 1, 2014• 10 citations
arXivHTMLPDF

Abstract

We provide a statistical communication model for the phenomenon of quantum information evaporation from black holes (BHs). A BH behaves as a reflecting quantum channel in a very special regime, which allows for a receiver to perfectly recover the absorbed quantum information. The quantum channel of a perfectly reflecting (PR) BH is the probabilistically weighted sum of infinitely many qubit cloning channels. In this work, we reveal the statistical communication background of the information evaporation process of PR BHs. We show that the density of the cloned quantum particles in function of the PR BH's mass approximates a Chi-square distribution, while the stimulated emission process is characterized by zero-mean, circular symmetric complex Gaussian random variables. The results lead to the existence of Rayleigh random distributed coefficients in the probability density evolution, which confirms the presence of Rayleigh fading (a special type of random fluctuation) in the statistical communication model of BH information evaporation.

DOI
10.1142/s0219749915600254

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We provide a statistical communication model for the phenomenon of quantum information evaporation from black holes (BHs). A BH behaves as a reflecting quantum channel in a very special regime, which allows for a receiver to perfectly recover the absorbed quantum information. The quantum channel of a perfectly reflecting (PR) BH is the probabilistically weighted sum of infinitely many qubit cloning channels. In this work, we reveal the statistical communication background of the information evaporation process of PR BHs. We show that the density of the cloned quantum particles in function of the PR BH's mass approximates a Chi-square distribution, while the stimulated emission process is characterized by zero-mean, circular symmetric complex Gaussian random variables. The results lead to the existence of Rayleigh random distributed coefficients in the probability density evolution, which confirms the presence of Rayleigh fading (a special type of random fluctuation) in the statistical communication model of BH information evaporation.
We study the properties of the quantum information transmission channel that emerges from the quantum dynamics of particles interacting with a black hole horizon. We calculate the quantum channel capacity in two limiting cases where a single-letter capacity is known to exist: the limit of perfectly reflecting and perfectly absorbing black holes. We find that the perfectly reflecting black hole channel is closely related to the Unruh channel and that its capacity is non-vanishing, allowing for the perfect reconstruction of quantum information outside of the black hole horizon. We also find that the complementary channel (transmitting entanglement behind the horizon) is entanglement-breaking in this case, with vanishing capacity. We then calculate the quantum capacity of the black hole channel in the limit of a perfectly absorbing black hole and find that this capacity vanishes, while the capacity of the complementary channel is non-vanishing instead. Rather than inviting a new crisis for quantum physics, this finding instead is in accordance with the quantum no-cloning theorem, because it guarantees that there are no space-like surfaces that contain both the sender’s quantum state and the receiver’s reconstructed quantum state.
We study the properties of the quantum information transmission channel that emerges from the quantum dynamics of particles interacting with a black hole horizon. We calculate the quantum channel capacity in two limiting cases where a single-letter capacity is known to exist: the limit of perfectly reflecting and perfectly absorbing black holes. We find that the perfectly reflecting black hole channel is closely related to the Unruh channel and that its capacity is non-vanishing, allowing for the perfect reconstruction of quantum information outside of the black hole horizon. We also find that the complementary channel (transmitting entanglement behind the horizon) is entanglement-breaking in this case, with vanishing capacity. We then calculate the quantum capacity of the black hole channel in the limit of a perfectly absorbing black hole and find that this capacity vanishes, while the capacity of the complementary channel is non-vanishing instead. Rather than inviting a new crisis for quantum physics, this finding instead is in accordance with the quantum no-cloning theorem, because it guarantees that there are no space-like surfaces that contain both the sender's quantum state and the receiver's reconstructed quantum state.
In this work, we describe optically inspired models for unitary black hole (BH) evaporation. The goal of these models are (i) to be operationally simple, (ii) approximately preserve the thermal nature of the emitted Hawking Radiation (HR), and (iii) attempt to reproduce the Page Curve that purports that information flows forth from the BH when it has evaporated to approximately half its initial mass. We concentrate on modeling the BH as a single mode squeezed state successively interacting, by means of beam splitters and squeezers, with vacuum modes near the horizon, giving rise to entangled pairs representing the external Hawking radiation and its partner particle inside the horizon. Since all states and operations are Gaussian throughout, we use a symplectic formalism to track the evolution of the composite system through the evolving means and variances of their quadrature operators. This allows us to easily compute correlations and entanglement between the BH and the HR, as well as calculate correlations between the BH at early and late times.
The black hole information paradox has been an important problem in quantum gravity. In the study of evaporating black hole, it has been proposed that the holographic map between the semiclassical effective description in bulk and the fundamental description in boundary cannot be isometric. In this work, we would like to study the reflected entropy in an evaporating black hole model through a nonisometric holographic map. We assume that the evaporating is slow enough that it makes sense to ascribe a slowly varying temperature to the Hawking radiation. We then introduce a two-sided black hole model to canonically purify the semiclassical state. The holographic map to the fundamental description is nonisometric and defined by a Haar random unitary matrix. We show that the entropy of radiation in the model computed in the fundamental description matches the result read from the quantum extremal surface formula and agrees with the Page curve. Furthermore, through a nonisometric holographic map, we study the reflected entropies between different regions, including the one between the black holes on different sides, the one between the radiations distributed symmetrically but disconnectedly, and the one between the black hole and the radiation on a single side. Our results are consistent with the existing ones based on the effective descriptions.
Black hole evaporation is one of the most striking phenomena at the interface between gravity and quantum physics. In Hawking's semi-classical treatment, where matter is quantum mechanical and the spacetime is definite and classical, evaporation leads to an apparent loss of unitarity of the overall evolution, and to the so-called black hole information paradox. Here, we go beyond this semi-classical treatment and formulate a toy quantum model of black hole evaporation that allows the black hole to evolve into a superposition of being fully evaporated and not fully evaporated, consistent with the Hawking particles being in a coherent superposition of different energy levels. We model Hawking particle production by the repeated action of quantum-controlled unitaries, generating emission from the quantum black hole and accounting for a quantum coherent back-reaction on the black hole matter state. We show that the probability of full annihilation of the black hole matter increases with time until the black hole is, asymptotically, fully evaporated in every branch of the quantum superposition. We prove that under natural assumptions, this evaporation model is unitary, such that the initial state can in principle be recovered from the final asymptotic state of the radiation.
Abstract In this paper we extend the investigation of Adami and Ver Steeg (2014 Class. Quantum Grav. 31 075015 ) to treat the process of black hole (BH) particle emission effectively as the analogous quantum optical process of parametric down conversion with a dynamical (depleted versus non-depleted) ‘pump’ source mode which models the evaporating BH energy degree of freedom. We investigate both the short time (non-depleted pump) and long time (depleted pump) regimes of the quantum state and its impact on the Holevo channel capacity for commu.nicating information from the far past to the far future in the presence of Hawking radiation. The new feature introduced in this work is the coupling of the emitted Hawking radiation modes through the common BH ‘source pump’ mode which phenomenologically represents a quantized energy degree of freedom of the gravitational field. This (zero-dimensional) model serves as a simplified arena to explore BH particle production/evaporation and back-action effects under an explicitly unitary evolution that enforces quantized energy/particle conservation. Within our analogous quantum optical model we examine the entanglement between two emitted particle/anti-particle and anti-particle/particle pairs coupled via the BH evaporating ‘pump’ source. We also analytically and dynamically verify the ‘Page information time’ for our model, which refers to the conventionally held belief that the information in the BH radiation becomes significant after the BH has evaporated half its initial energy into the outgoing radiation. Lastly, we investigate the effect of BH particle production/evaporation on two modes in the exterior region of the BH event horizon that are initially maximally entangled, when one mode falls inward and interacts with the BH, and the other remains forever outside and non-interacting.