Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Entropy

Population–Coherence Routes to Purity in Page-Type Models of Black-Hole Evaporation

José J. Gil

Published Feb 27, 2026• 0 citations
Publisher

Abstract

-dimensional 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
10.3390/e28030263

Related papers

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

José J. Gil

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.

arXiv:2602.18503•Feb 17, 2026•0 citations

An algebraic description of the Page transition

Haocheng Zhong

In this work, we develop an algebraic description of the Page transition, a key feature in black hole evaporation where the entropy of Hawking radiation follows a unitary Page curve instead of monotonically increasing. By applying concepts from approximate quantum error correction with complementary recovery, we characterize the Page transition as a phase transition in channel recovery. We then generalize the description to infinite-dimensional settings using algebraic relative entropy, which remains valid even in type III factors. For type I/II factors, explicit probes based on relative entropy differences are derived, serving as indicators for the transition at the Page time.

arXiv:2601.11363•Jan 16, 2026•0 citations

A two-mode model for black hole evaporation and information flow

Erfan Bayenat, Babak Vakili

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 $\langle n_x\rangle - \langle n_y\rangle$. The reduced entropy $S_x(t)$ 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.

quant-pharXiv:2601.19737•Jan 28, 2026

An algebraic description of the Page transition

Haocheng Zhong

A bstract In this work, we develop an algebraic description of the Page transition, a key feature in black hole evaporation where the entropy of Hawking radiation follows a unitary Page curve instead of monotonically increasing. By applying concepts from approximate quantum error correction with complementary recovery, we characterize the Page transition as a phase transition in channel recovery. We then generalize the description to infinite-dimensional settings using algebraic relative entropy, which remains valid even in type III factors. For type I/II factors, explicit probes based on relative entropy differences are derived, serving as indicators for the transition at the Page time.

Journal of High Energy Physics•Apr 20, 2026•0 citations

Energy Records and Relational Clocks: An Analytic Decomposition in a Black-Hole–Radiation Toy Model

Jérôme Turcotte

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.

•Sep 7, 2026•0 citations

Conditional entanglement transfer via black holes: restoring predictability

Ali Akil, Oscar Dahlsten, Leonardo Modesto

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.

New Journal of Physics•Jul 26, 2021•2 citations
-dimensional 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.
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.
In this work, we develop an algebraic description of the Page transition, a key feature in black hole evaporation where the entropy of Hawking radiation follows a unitary Page curve instead of monotonically increasing. By applying concepts from approximate quantum error correction with complementary recovery, we characterize the Page transition as a phase transition in channel recovery. We then generalize the description to infinite-dimensional settings using algebraic relative entropy, which remains valid even in type III factors. For type I/II factors, explicit probes based on relative entropy differences are derived, serving as indicators for the transition at the Page time.
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.
A bstract In this work, we develop an algebraic description of the Page transition, a key feature in black hole evaporation where the entropy of Hawking radiation follows a unitary Page curve instead of monotonically increasing. By applying concepts from approximate quantum error correction with complementary recovery, we characterize the Page transition as a phase transition in channel recovery. We then generalize the description to infinite-dimensional settings using algebraic relative entropy, which remains valid even in type III factors. For type I/II factors, explicit probes based on relative entropy differences are derived, serving as indicators for the transition at the Page time.
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.
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.