Qubit Models of Black Hole Evaporation

TL;DR

This work develops a general qubit-based framework to compare unitary and nonunitary black hole evaporation within the language of quantum operations. By embedding the evolution in an enlarged Hilbert space and detailing a two-step pair-creation plus internal-dynamics structure, it clarifies how small corrections fail to restore unitarity and illustrates that substantial, nonlocal, or horizon-scale corrections are required. Through a spectrum of explicit models and a rigorous generalization of Mathur's bound, the paper shows that unitarity can be approached only via large deformations from Hawking's semiclassical evolution, and it provides a concrete one-parameter interpolation demonstrating the distance between Hawking radiation and fully unitary evaporation. The results offer a unifying, information-theoretic perspective on the role of auxiliary degrees of freedom and the kinds of corrections necessary to reconcile black hole evaporation with quantum mechanics, with implications for fuzzball and nonlocality scenarios.

Abstract

Recently, several simple quantum mechanical toy models of black hole evaporation have appeared in the literature attempting to illuminate the black hole information paradox. We present a general class of models that is large enough to describe both unitary and nonunitary evaporation, and study a few specific examples to clarify some potential confusions regarding recent results. We also generalize Mathur's bound on small corrections to black hole dynamics. Conclusions are then drawn about the requirements for unitary evaporation of black holes in this class of models. We present a one-parameter family of models that continuously deforms nonunitary Hawking evaporation into a unitary process. The required deformation is large.

Figures (2)

• Figure 1: Here we illustrate the black hole evolution with auxillary degrees of freedom. At first we have only the initial black hole degrees of freedom represented by dots. At later stages in the evolution, we introduce some new degrees of freedom that corresponding to the infalling negative energy particles in the Hawking process (squares), as well as outgoing radiation (wavy lines). We have thus increased the size of the Hilbert space, but this should be thought of only as a convenient way to parametrize potentially nonunitary evolution. To return to a fixed dimension model, one needs to trace out auxillary degrees of freedom. In general, at intermediate steps, it is not clear which degrees of freedom should be thought of as auxillary, so we illustrate just one possibility. On the other hand, it is unambiguous that in the final state all of the black hole degrees of freedom (dots and squares) are auxillary if there are no remnants.
• Figure 2: Here we present the second Rényi entropies of the unhatted radiation qubits as a function of $\theta$ for the model in Equation \ref{['eq:theta-model']} with three different initial states. Intermediate steps are dashed curves and the final step is solid; the steps are shown in the color order (red, yellow, green, blue, purple, black). The point $\theta=0$ corresponds to the canonical Hawking evolution, while $\theta=\frac{\pi}{2}$ corresponds to the unitary model in Equation \ref{['eq:G2']}.