A Quantum Circuit Model of Black Hole Evaporation with Tunable Semi-Causality Violation
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 $\mathrm{CU}(\sigma)$ 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 $\sigma=0$, the model reproduces Broda's Page-like entropy evolution with complete late-time purification. For any $\sigma>0$, 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-$\sigma$ regime, the residual entropy exhibits a characteristic $-\sigma^{2}\ln\sigma^{2}$ scaling that bears qualitative similarity to logarithmic entropy corrections in generalized uncertainty principle inspired evaporation scenarios. For larger values of $\sigma$, 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.