We formulate quantum evolution on growing causal domains as a family of quantum channels between generally non-isomorphic state spaces. Unitarity is replaced locally by approximate isometric evolution, measured with an energy-constrained diamond norm, while observer covariance is imposed through consistency, refinement, and cocycle conditions on causal overlaps. Applied to an evaporating black hole, the formalism yields a discrete isometric model in which information is transferred from the black-hole sector to the radiation sector without requiring a fixed global Hilbert-space factorization. A random-isometry approximation reproduces the Page transition and shows that a dynamically populated complementary sector produces a calculable endpoint defect in the Page curve. Information--disturbance bounds further imply that exact exterior unitarity forces the complementary channel to be input independent, while approximate unitarity quantitatively bounds structural leakage. The resulting framework interprets unitarity as an emergent property of observer-consistent causal descriptions rather than as a fundamental identification of state spaces at different times.
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) 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
−σ2lnσ2 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.
We present a minimal, fully unitary toy model of black hole evaporation in which information recovery emerges from non-factorized global states and observer-dependent access. The model reproduces a Page-like entropy curve for Hawking radiation without invoking fundamental information loss, horizon microstructure, or spacetime geometry at the microscopic level. The result supports the interpretation that the black hole information paradox arises from regime-dependent coarse-graining rather than from a breakdown of quantum unitarity.
This paper presents a structural resolution of the black-hole information-loss contradiction using standard quantum-information and operator-algebraic ideas. The central argument is that local horizon regularity, unitary evolution, locally thermal-looking radiation, no cloning, and complete final information recovery are not mutually inconsistent unless one additionally assumes that the interior partner and the radiation reconstruction must remain permanently independent physical subsystems. The paper develops an alternative framework in which globally encoded quantum information can remain invisible to the local horizon observable algebra while being preserved exactly in the total quantum state and progressively transferred into correlations within the emitted radiation. An explicit finite-dimensional code model demonstrates that each individual emitted subsystem can appear locally universal while arbitrary quantum information, including entanglement with an external reference system, remains exactly preserved and fully recoverable from the final radiation. The analysis also distinguishes the local horizon-partner role from the global microstate information and from the physical carrier that realizes either role at a given stage of evaporation. This separation removes the apparent monogamy conflict without violating linearity, no cloning, or unitary quantum mechanics. A complementary entropy analysis shows how a Page-like transition can arise when the responsibility for purification becomes progressively reconstructible from the radiation. The paper does not claim a complete dynamical theory of astrophysical black-hole evaporation. Instead, it establishes a mathematically explicit consistency framework showing that the standard information-loss contradiction is not forced by local thermality and unitarity alone, and it identifies the remaining physical problem as the derivation of the appropriate horizon observable algebra and evaporation dynamics from a complete theory of gravity.
We construct a toy model of an evaporating black hole using approximately local degrees of freedom acting on ``overlapping" qubits in which a version of black hole complementarity arises naturally. The operators corresponding to the radiation and the interior are identified as two distinct representations of the same fundamental algebra, thereby preventing the exact factorization of the Hilbert space into interior and exterior and avoiding the conventional no-cloning violations. We show how this toy model captures several qualitative and quantitative features of black hole evaporation and how the ability to account for this ``overlap" in the entropy calculation leads to the recovery of a Page curve.
Abstract Black-hole evaporation can be unitary only if Hawking's thermal spectrum is a coarse view of an underlying pure quantum process. Island and replica-wormhole calculations recover the Page curve but do not generally provide the amplitudes mapping a specified black-hole microstate to outgoing quanta. Here we construct such a state-resolved channel for an area-law quantum memory in matrix black-hole quantum mechanics. An exact symmetry separates the fermionic state into a geometric factor that controls emission and a protected factor untouched by the full real- and imaginary-time matrix and link dynamics. We prove that this memory occurs in every representation sector, enumerate all sectors through rank ten, and exhibit a self-complementary sector whose entropy scales with horizon area. When the matrix rank decreases, an exact branching isometry transfers the memory into daughter-hole and radiation degrees of freedom. Combining this isometry with the model's monopole-tunnelling weights gives every encoded state the same Hawking-thermal branch probabilities while its conditional radiation remains pure and microstate dependent. Finite-rank simulations show Page-like transfer and near-unit decoding; dephasing leaves the branch probabilities unchanged but erases recovery. The result provides an explicit amplitude-level mechanism by which Hawking thermality and unitary information transfer coexist in one microscopic model.
Almheiri et al. have emphasized that otherwise reasonable beliefs about black hole evaporation are incompatible with the monogamy of quantum entanglement, a general property of quantum mechanics. We investigate the final-state projection model of black hole evaporation proposed by Horowitz and Maldacena, pointing out that this model admits cloning of quantum states and polygamous entanglement, allowing unitarity of the evaporation process to be reconciled with smoothness of the black hole event horizon. Though the model seems to require carefully tuned dynamics to ensure exact unitarity of the black hole S-matrix, for a generic final-state boundary condition the deviations from unitarity are exponentially small in the black hole entropy; furthermore observers inside black holes need not detect any deviations from standard quantum mechanics. Though measurements performed inside old black holes could potentially produce causality-violating phenomena, the computational complexity of decoding the Hawking radiation may render the causality violation unobservable. Final-state projection models illustrate how inviolable principles of standard quantum mechanics might be circumvented in a theory of quantum gravity.