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arXiv:2602.18503

Population-coherence routes to purity in Page-type models of black-hole evaporation

José J. Gil

Published Feb 17, 2026• 0 citations
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Abstract

We revisit the black-hole information problem from the viewpoint of a population-coherence decomposition of density-matrix purity. Building on a previously developed formalism for ndimensional density matrices, we characterize each state by a normalized global purity index and two complementary indices, which quantify the contributions of level populations and coherences. This yields a simple quadratic relation and a geometric representation in a population-coherence plane, where different routes to purity can be distinguished. In the two-level case we construct explicit families of states with identical spectra and global purity but opposite internal structure, realizing population-dominated and coherence-dominated routes. We then apply this framework to a standard Page-type evaporation model without an explicit Hamiltonian, in which a black hole and its Hawking radiation form a bipartite pure state with varying Hilbert-space dimensions. Using known results for typical reduced states in large dimensions, we analyze the behavior of population and coherence components of purity along the evaporation process. Under the physically motivated requirement that, in this energy-free setting, the radiation populations remain nearly uniform in the chosen basis, we show that the late-time recovery of purity must be coherence-dominated: the global purity of the radiation approaches unity while the population index stays small and the coherence index carries essentially all the purity.

DOI
10.48550/arxiv.2602.18503

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We revisit the black-hole information problem from the viewpoint of a population-coherence decomposition of density-matrix purity. Building on a previously developed formalism for ndimensional density matrices, we characterize each state by a normalized global purity index and two complementary indices, which quantify the contributions of level populations and coherences. This yields a simple quadratic relation and a geometric representation in a population-coherence plane, where different routes to purity can be distinguished. In the two-level case we construct explicit families of states with identical spectra and global purity but opposite internal structure, realizing population-dominated and coherence-dominated routes. We then apply this framework to a standard Page-type evaporation model without an explicit Hamiltonian, in which a black hole and its Hawking radiation form a bipartite pure state with varying Hilbert-space dimensions. Using known results for typical reduced states in large dimensions, we analyze the behavior of population and coherence components of purity along the evaporation process. Under the physically motivated requirement that, in this energy-free setting, the radiation populations remain nearly uniform in the chosen basis, we show that the late-time recovery of purity must be coherence-dominated: the global purity of the radiation approaches unity while the population index stays small and the coherence index carries essentially all the purity.
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We study a stationary finite-dimensional bipartite quantum system composed of mass levels, labelled “black hole,” and energy levels, labelled “radiation.” Matched levels satisfy a fixed total-energy constraint. Analytically and across 128 paired numerical realizations, the diagonal part of the global state provides a perfect record between remaining mass and radiated energy, whereas off-diagonal coherence across energy branches supports Page–Wootters conditional evolution. Complete dephasing preserves stationarity, the energy shell, both imposed thermal local density matrices, and the classical energy record, while eliminating all nontrivial clock variation. Relational clocks are detected in 128/128 coherent states and 0/384 mixed controls. Within this finite family, diagonal correlations encode the record of change, whereas coherence supports relational change. This construction is a controlled toy model. It does not reproduce black-hole geometry, Hawking pair creation, backreaction, or the Page curve, and it does not claim to resolve the black-hole information paradox.
We develop and analyze a two-oscillator model for black hole evaporation in which an effective geometric degree of freedom and a representative Hawking radiation mode are described by coupled harmonic oscillators with opposite signs in their free Hamiltonians. The normal-mode structure is obtained analytically and the corresponding modal amplitudes determine the pattern of energy exchange between the two sectors. To bridge the discrete and semiclassical pictures, we introduce smooth envelope functions that provide a continuous effective description along the geometric variable. Numerical simulations in a truncated Fock space show that the two oscillators exchange quanta in an approximately out-of-phase manner, consistent with an effective conservation of ⟨nx⟩−⟨ny⟩\langle n_x\rangle - \langle n_y\rangle⟨nx​⟩−⟨ny​⟩. The reduced entropy Sx(t)S_x(t)Sx​(t) exhibits periodic growth, indicating entanglement generation. These results demonstrate that even a minimal two-mode framework can capture key qualitative features of energy transfer and information flow during evaporation.
Hawking's black hole evaporation process suggests that we may need to choose between quantum unitarity and other basic physical principles such as no-signaling, entanglement monogamy, and the equivalence principle. We here show that the Hawking's quantum model for the black hole evaporation is consistent with the above fundamental principles. Our analysis does not involve exotic new physics, but rather uses standard quantum theory, general relativity, and the Einstein–Hilbert action including matter. We explicitly show that the whole state consisting of matter and radiation (in a joint superposition of different energy states) is pure at any stage of the evaporation process, including the particular case of 0 mass. Moreover, after full evaporation the state for radiation at infinity is pure and in one-to-one correspondence with the initial state forming the black hole. Thus there is no information loss upon full evaporation according to the quantum information theory. The original entanglement of the black hole matter (if any) gets transferred to the outgoing particles via a process similar to entanglement swapping, without violation of causality (as proved explicitly). On the other hand, if the initial state is a tensor product state, the entanglement of Hawking particles, present in the intermediate phase, is broken when the black hole evaporates completely. Therefore, the final state (entangled or tensor product depending on the nature of initial state) after the full black hole evaporation is pure without loss of information.
Abstract We present an analytical formulation of the recent one-shot decoupling model of Bràdler and Adami (2015 arXiv: 1505.0284 ) and compute the resulting ‘Page information’ curves, for the reduced density matrices for the evaporating black hole (BH) internal degrees of freedom, and emitted Hawking radiation pairs entangled across the horizon. We argue that BH evaporation/particle production has a very close analogy to the laboratory process of spontaneous parametric down conversion, when the pump is allowed to deplete.
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.