Entanglement and firewalls in quantum circuit model of black hole evaporation

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 ( $$\textbf{BH}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>BH</mml:mi> </mml:math> ), just radiation ( $$\textbf{JR}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>JR</mml:mi> </mml:math> ), and early radiation ( $$\textbf{ER}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ER</mml:mi> </mml:math> ). We apply a scrambling unitary matrix with a single parameter $$\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> 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 $$\theta =0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and $$\theta =\pi /2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>π</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , 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 $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> . For the initial state with $$\theta =\pi /2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>π</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , a firewall emerges at an earlier stage of the evolution than with $$\theta =0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . We also find that a firewall structure emerges between $$\textbf{BH}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>BH</mml:mi> </mml:math> and $$\textbf{JR}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>JR</mml:mi> </mml:math> , and that the information is carried away by radiation for all values of $$M\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mi>ω</mml:mi> </mml:mrow> </mml:math> , provided that $$\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> 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.