Quantum Circuit Model of Black Hole Evaporation with Controlled Causal Leakage

Abstract

We extend a four-qubit quantum circuit model of black hole evaporation that enforces semi-causality, a condition that allows information to enter a black hole but strictly forbids any information from escaping from the interior to outside through the horizon. In this work, we introduce a controlled violation of this principle by inserting a parametric controlled-unitary gate that enables a tunable leakage of quantum information from the black hole interior to the exterior, while preserving global unitarity. By varying the deformation parameter, we study the evolution of entanglement entropy, mutual information, and entanglement negativity throughout the evaporation process. While the semi-causal case yields a Page-like entropy curve with vanishing late-time correlations, we find that even small violations of semi-causality produce a non-zero residual entropy and persistent negativity across the horizon. These features mimic quantum-gravity-induced effects such as remnant formation and horizon permeability, suggesting that minimal deviations from classical causality can leave long-lived imprints on black hole information dynamics.

Figures (5)

• Figure 1: Pictorial representation of the qubit model for black hole evaporation. The four qubits correspond to BH (interior), GR (near-horizon gravity), IN (infalling partner), and OUT (Hawking radiation).
• Figure 2: Quantum circuit for black hole evaporation with tunable semi-causality violation. The event horizon is indicated by a double grey line with thicker line corresponding to the interior of the black hole. The parametric controlled-unitary gates $\mathrm{CU}(\sigma)$ introduce controlled leakage across the horizon and within the Hawking pair.
• Figure 3: Time evolution of the von Neumann entropy $S(\tau)$ for the BH and OUT subsystems at different values of the semi-causality violation parameter $\sigma$. For $\sigma = 0$, the entropy follows a standard Page curve and vanishes after the Page time. For $\sigma > 0$, the entropy saturates to a non-zero value, indicating persistent entanglement and incomplete information retrieval. We set the input parameters as $|\theta|= |\eta| = 1/\sqrt{2}$ and $|\alpha| = |\beta| = 1/\sqrt{2}$.
• Figure 4: Numerical evolution of entanglement measures across time steps $\tau$ for various values of $\sigma$. Each row corresponds to a different measure (entropy, mutual information, negativity), and the columns separate the bi-partitions: BH-GR and IN-OUT. For $\sigma > 0$, all quantities exhibit non-zero late-time values, signalling persistent correlations across the event horizon. The parameters are chosen such that $|\theta|= |\eta| = 1/\sqrt{2}$ and $|\alpha| = |\beta| = 1/\sqrt{2}$).
• Figure 5: Residual entropy $S_{\mathrm{BH}}(\tau_4)$ of the black hole subsystem as a function of the semi-causality violation parameter $\sigma$, for three choices of initial pair entanglement $|\eta| \in \{0.25,\; 0.5,\; 1/\sqrt{2}\}$. The residual entropy increases with $\sigma$, indicating persistent entanglement across the horizon. Lower values of $|\eta|$ lead to smaller entropy, consistent with reduced initial correlations in the Hawking pair. This complements the Page-curve analysis in Fig. \ref{['fig:entropy_pagecurve']}, offering a parametric view of late-time information retention.