We present a four-qubit quantum circuit model of black hole evaporation with a controlled violation of semi-causality, understood as the condition that information may fall into the black hole but cannot propagate back across the horizon. Building on Broda's semi-causal evaporation circuit, we introduce a controlled-unitary gate
CU(σ) that allows tunable information leakage from the interior to the exterior while preserving global unitarity.
We compute the single-qubit reduced entropies, together with the mutual information and entanglement negativity for the BH-GR and IN-OUT bipartitions, at each discrete time step. For
σ=0, the model reproduces Broda's Page-like entropy evolution with complete late-time purification. For any
σ>0, however, nonzero residual single-qubit entropies and persistent late-time entanglement remain, indicating incomplete purification of the outgoing radiation despite the global unitary evolution.
In the small-
σ regime, the residual entropy exhibits a characteristic
−σ2lnσ2 scaling that bears qualitative similarity to logarithmic entropy corrections in generalized uncertainty principle inspired evaporation scenarios. For larger values of
σ, the persistence of finite residual entropy invites comparison with remnant-like endpoint configurations in regular or extremal black hole models. Our results show how controlled departures from the semi-causal limit modify information recovery in a minimal analytically tractable model of black hole evaporation.
We propose a framework for understanding the information loss paradox in the context of canonical quantum gravity. We first revisit several approaches to black hole evaporation that do not involve an event horizon. In these models, we assume that each time slice corresponds to a coherent state in the gravitational phase space. Due to this property, the quantum state associated with a given spatial configuration will have a nonzero overlap with another spatial configuration. We show how this property can account for both semi-classical dynamics and unitary time evolution. We introduce an explicit toy model that follows semi-classical time evolution at the coarse-grained level; in addition, its time evolution is unitary at the fine-grained level. The only price that one must pay is the violation of the entropy bound; we discuss reasons why this may be justifiable.
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 We reexamine the quantum circuit model of black hole evaporation proposed in a previous work (Class. Quantum Grav. 35, 235013, 2018) [1]. This tripartite model incorporates the following systems: black hole (
BH BH ), just radiation (
JR JR ), and early radiation (
ER ER ). We apply a scrambling unitary matrix with a single parameter
θ θ to the ground state of the qubits in infalling matter toward a black hole in order to generate initial qubit states of the black hole that are more general than those in [1]. Specifically, the scrambling unitary matrix reduces to no scrambling and maximum scrambling when
θ=0 θ = 0 and
θ=π/2 θ = π / 2 , respectively. Our aim is to explore the role of quantum monogamy in the firewall formation between the black hole and radiation. In this model, entanglement and firewall formation depend on the black hole mass M and the frequency of Hawking radiation
ω ω . For the initial state with
θ=π/2 θ = π / 2 , a firewall emerges at an earlier stage of the evolution than with
θ=0 θ = 0 . We also find that a firewall structure emerges between
BH BH and
JR JR , and that the information is carried away by radiation for all values of
Mω M ω , provided that
θ θ lies within a certain analytically determined range. Following unitary gate dynamics, the initial black hole qubit state can be retrieved from its imprint on the final radiation state, which was originally hidden behind the black hole’s horizon. These results may provide insight into the properties of multipartite entanglement due to the different initial states in the evolution of a quantum circuit model for black hole evaporation.
Consider a bipartite entangled system half of which falls through the event horizon of an evaporating black hole, while the other half remains coherently accessible to experiments in the exterior region. Beyond complete evaporation, the evolution of the quantum state past the Cauchy horizon cannot remain unitary, raising the questions: How can this evolution be described as a quantum map, and how is causality preserved? What are the possible effects of such nonstandard quantum evolution maps on the behavior of the entangled laboratory partner? More generally, the laws of quantum evolution under extreme conditions in remote regions (not just in evaporating black-hole interiors, but possibly near other naked singularities and regions of extreme spacetime structure) remain untested by observation, and might conceivably be non-unitary or even nonlinear, raising the same questions about the evolution of entangled states. The answers to these questions are subtle, and are linked in unexpected ways to the fundamental laws of quantum mechanics. We show that terrestrial experiments can be designed to probe and constrain exactly how the laws of quantum evolution might be altered, either by black-hole evaporation, or by other extreme processes in remote regions possibly governed by unknown physics.