A simple model of entangled qubits: how it describes superconductors and black holes

L. Borsten

Bogusław Broda

A new simple qubit model of black hole evaporation is proposed. The model operates on four qubits and is defined in terms of quantum gates and a quantum circuit. The chief features of the model include explicit unitarity and (most notably) causality which is understood as the impossibility of the transfer of information from the interior of the black hole through its horizon. The corresponding von Neumann entanglement entropy yields a crude (four-qubit) approximation of the Page curve.

Sophia E. Economou

Sourav Ballav, Wen-Yu Wen, Chi-Hsien Tai

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 $\mathrm{CU}(σ)$ 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 $-σ^{2}\lnσ^{2}$ 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.

Y S Lü, Tien Hsieh, Da-Shin Lee

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.

A.‐B. A. Mohamed, Hichem Eleuch