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arXiv:2606.21181 · The European Physical Journal C

Entanglement and firewalls in quantum circuit model of black hole evaporation

Y S Lü, Tien Hsieh, Da-Shin Lee

Published Jul 26, 2026• 0 citations
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Abstract

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 ( $$\textbf{BH}$$ BH ), just radiation ( $$\textbf{JR}$$ JR ), and early radiation ( $$\textbf{ER}$$ ER ). We apply a scrambling unitary matrix with a single parameter $$\theta $$ 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 $$\theta =0$$ θ = 0 and $$\theta =\pi /2$$ 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 $$\omega $$ ω . For the initial state with $$\theta =\pi /2$$ θ = π / 2 , a firewall emerges at an earlier stage of the evolution than with $$\theta =0$$ θ = 0 . We also find that a firewall structure emerges between $$\textbf{BH}$$ BH and $$\textbf{JR}$$ JR , and that the information is carried away by radiation for all values of $$M\omega $$ M ω , provided that $$\theta $$ 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.

DOI
10.1140/epjc/s10052-026-16001-7

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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\textbf{BH}BH BH ), just radiation ( JR\textbf{JR}JR JR ), and early radiation ( ER\textbf{ER}ER ER ). We apply a scrambling unitary matrix with a single parameter θ\theta 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\theta =0θ=0 θ = 0 and θ=π/2\theta =\pi /2θ=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\omega ω ω . For the initial state with θ=π/2\theta =\pi /2θ=π/2 θ = π / 2 , a firewall emerges at an earlier stage of the evolution than with θ=0\theta =0θ=0 θ = 0 . We also find that a firewall structure emerges between BH\textbf{BH}BH BH and JR\textbf{JR}JR JR , and that the information is carried away by radiation for all values of MωM\omega Mω M ω , provided that θ\theta 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.
Abstract We consider a quantum circuit model describing the evaporation process of black holes. We specifically examine the behavior of the multipartite entanglement represented by this model, and find that the entanglement structure depends on the black hole mass M and the frequency of the Hawking radiation ω . For sufficiently small values of , the black hole and the radiation system becomes a separable state after the Page time and a firewall-like structure appears. On the contrary, for larger values of , the entanglement between the black hole and the radiation is not lost. These behaviors imply that owing to the monogamy property of the multipartite entanglement, low frequency modes of the Hawking radiation destroy the quantum correlation between the black hole and the emitted Hawking radiation.
We suggest a quantum circuit model which simulates the black-hole evaporation process. In particular, Almheiri-Marolf-Polchinski-Sully (AMPS) paradox and the ER=EPR correspondence are reconsidered regarding our proposed model, which assumes a Maxwell's demon operating within a black hole interior. In other words, we form a quantum circuit, mimicking the behavior of the entanglement structure of the near-horizon region and the early Hawking radiation located far from the black hole. Furthermore, we indicate how the demon, by applying nonlocal correlations, can mediate via Einstein-Rosen bridges for the purpose of simulating the transfer of quantum information across the horizon without violating the monogamy of entanglement. Finally, the thermodynamic cost of the demon's operations regarding Landauer's principle is analyzed. This indicates that the information erasure has an energy comparable to the black hole entropy.
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)\mathrm{CU}(σ)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=0σ=0σ=0, the model reproduces Broda's Page-like entropy evolution with complete late-time purification. For any σ>0σ>0σ>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-σσσ regime, the residual entropy exhibits a characteristic −σ2ln⁡σ2-σ^{2}\lnσ^{2}−σ2lnσ2 scaling that bears qualitative similarity to logarithmic entropy corrections in generalized uncertainty principle inspired evaporation scenarios. For larger values of σσσ, 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.
Hawking's black hole evaporation process suggests that we may need to choose between quantum unitarity and other basic physical principles such as no-signalling, entanglement monogamy, and the equivalence principle. We here provide a quantum model for Hawking pair black hole evaporation within which these principles are all respected. The model does not involve exotic new physics, but rather uses quantum theory and general relativity. The black hole and radiation are in a joint superposition of different energy states at any stage of the evaporation process. In the particular branch where the black hole mass is 0, the radiation state is pure and one-to-one with the initial state forming the black hole. Thus there is no information loss upon full evaporation. The original Hawking's pair entanglement between infalling and outgoing particles gets transferred to outgoing particles via entanglement swapping, without violation of no-signalling or the entanglement's monogamy. The final state after the full black hole evaporation is pure, without loss of information, violation of monogamy, or the equivalence principle.
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.
Hawking's black hole evaporation process suggests that we may need to choose between quantum unitarity and other basic physical principles such as no-signaling, entanglement monogamy, and the equivalence principle. We here show that the Hawking's quantum model for the black hole evaporation is consistent with the above fundamental principles. Our analysis does not involve exotic new physics, but rather uses standard quantum theory, general relativity, and the Einstein–Hilbert action including matter. We explicitly show that the whole state consisting of matter and radiation (in a joint superposition of different energy states) is pure at any stage of the evaporation process, including the particular case of 0 mass. Moreover, after full evaporation the state for radiation at infinity is pure and in one-to-one correspondence with the initial state forming the black hole. Thus there is no information loss upon full evaporation according to the quantum information theory. The original entanglement of the black hole matter (if any) gets transferred to the outgoing particles via a process similar to entanglement swapping, without violation of causality (as proved explicitly). On the other hand, if the initial state is a tensor product state, the entanglement of Hawking particles, present in the intermediate phase, is broken when the black hole evaporates completely. Therefore, the final state (entangled or tensor product depending on the nature of initial state) after the full black hole evaporation is pure without loss of information.