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Information Flow from Zero-Energy Boundary Modes: Exact Quantum-Nondemolition Dynamics of Soft Hair in a Solvable Gauge Model

Mustafa Boudjerada

Published Aug 25, 2026• 0 citations
Publisher

Abstract

The black-hole soft-hair proposal raises a central dynamical question: can zero-energy boundary degrees of freedom become correlated with radiation in a way that preserves and reveals information, or are such modes permanently decoupled from observable physics? This work studies that question in an exactly solvable minimal charge-locked emission model derived from the soft-hair structure of the Schwinger model. The model provides a controlled setting to analyze information transfer between boundary hair degrees of freedom and emitted radiation without uncontrolled approximations. The analysis shows that the hair variable acts as a quantum-nondemolition degree of freedom: the hair charge is conserved during emission, while the radiation acquires a coherent record of the initial sector. The emitted radiation remains phase-locked to the vacuum-sector label, and the decay of hair coherences follows an exact closed-form decoherence law. The information flow is characterized using mutual information and explicit state reconstruction. The radiation record approaches a perfect distinguishable record of the hair sector, and pretty-good-measurement decoding achieves exponentially improving recovery accuracy, reaching errors below 10⁻¹² in the verified regime. A central result is an operator-class separation theorem within the model: neutral radiation observables remain completely insensitive to the hair information at all times, while charged observables can access the encoded phase information. Thus, the question of whether soft hair carries information is resolved in this solvable setting by distinguishing the operator algebra available to the observer. The result provides an exactly computable benchmark for discussions of soft hair, information transfer, superselection sectors, and quantum reference frames. The work does not claim a black-hole evaporation solution: the model has no gravitational backreaction, no Page-curve dynamics, and is formulated in a 1+1-dimensional gauge-theory setting. All analytic results, numerical checks, and verification procedures are accompanied by reproducibility materials.

DOI
10.5281/zenodo.22088379

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The black-hole soft-hair proposal raises a central dynamical question: can zero-energy boundary degrees of freedom become correlated with radiation in a way that preserves and reveals information, or are such modes permanently decoupled from observable physics? This work studies that question in an exactly solvable minimal charge-locked emission model derived from the soft-hair structure of the Schwinger model. The model provides a controlled setting to analyze information transfer between boundary hair degrees of freedom and emitted radiation without uncontrolled approximations. The analysis shows that the hair variable acts as a quantum-nondemolition degree of freedom: the hair charge is conserved during emission, while the radiation acquires a coherent record of the initial sector. The emitted radiation remains phase-locked to the vacuum-sector label, and the decay of hair coherences follows an exact closed-form decoherence law. The information flow is characterized using mutual information and explicit state reconstruction. The radiation record approaches a perfect distinguishable record of the hair sector, and pretty-good-measurement decoding achieves exponentially improving recovery accuracy, reaching errors below 10⁻¹² in the verified regime. A central result is an operator-class separation theorem within the model: neutral radiation observables remain completely insensitive to the hair information at all times, while charged observables can access the encoded phase information. Thus, the question of whether soft hair carries information is resolved in this solvable setting by distinguishing the operator algebra available to the observer. The result provides an exactly computable benchmark for discussions of soft hair, information transfer, superselection sectors, and quantum reference frames. The work does not claim a black-hole evaporation solution: the model has no gravitational backreaction, no Page-curve dynamics, and is formulated in a 1+1-dimensional gauge-theory setting. All analytic results, numerical checks, and verification procedures are accompanied by reproducibility materials.
The information content of quantum field theory states depends on which operator algebra is available to an observer. Casini, Huerta, Magán, and Pontello developed entropic order parameters based on relative entropy to quantify which algebras can access superselection-sector information and established entropic complementarity relations for complementary algebra choices. This work computes that framework exactly in the θ-vacuum manifold of the massless Schwinger model, providing a fully solvable quantum field theory realization of the entropic structure associated with sector information. The neutral-algebra restriction of a θ-state is shown exactly to be the uniform mixture of flux sectors and completely independent of θ. Consequently, the relative entropy between different θ-sectors on the neutral algebra vanishes: the neutral algebra carries no information about the vacuum angle. The entropic order parameter is computed in closed form. For equal sector weights, the relative entropy order parameter is exactly ln(N), while for general sector weights it becomes the Shannon entropy H(p) of the sector distribution. The conjugate pinching in the θ-basis provides the complementary disorder parameter, and the pair realizes the entropic complementarity relation, including saturation of the Maassen–Uffink bound at the extreme limits. Combining these results with the companion vacuum decomposition results gives an exact information-theoretic realization of the principle in this model: the θ-sector information is globally present and perfectly distinguishable, absent from the neutral algebra, absent from null-surface data within the stated universality framework, and carried by the charged/boundary sector while requiring no energy cost. This work provides a fully solvable benchmark for entropic superselection frameworks in quantum field theory. The declared limits are explicit: the calculation is performed for a finite sector structure in the 1+1 dimensional Schwinger model; continuum algebraic results are cited rather than re-derived; no universal theorem and no gravitational claim are made. All symbolic derivations and numerical checks are accompanied by reproducibility materials, verification scripts, audit procedures, and recorded outputs.
The Page curve is the standard signature of unitary information recovery in an evaporating quantum system. This work asks a question usually left implicit: whose Page curve? In a solvable charge-locked evaporation model, the radiation observables separate into two classes: the charge-coherent algebra, available to an observer with the appropriate charge reference frame, and the charge-invariant algebra of an observer without such a reference frame. These two observers do not recover the same information curve. For the full charge-coherent algebra, the Page curve follows the standard unitary pattern: the radiation entropy rises and then returns exactly to zero, with all interior microstates becoming mutually orthogonal and perfectly decodable radiation records. For the charge-invariant algebra, the turnover is incomplete. The accessible entropy terminates at the charge-distribution entropy, representing the information deficit caused by the missing reference frame. Microstates related by a global charge-phase transformation remain indistinguishable to this observer, not because information is destroyed, but because it is inaccessible within that observable algebra. The work establishes an operator-class Page theorem: complete purification and full information recovery are properties of the charge-coherent algebra, while charge-invariant observers experience a Page curve truncated by the entropy associated with the missing charge reference frame. The model also reveals a mode-structure law. Multi-channel charged radiation can carry relative-charge information internally and locks only the global charge-phase bit, while a single effective channel can lock the full information content. An external reference frame restores accessibility only when its charge quantum matches the charge quantum of the hidden coherence, demonstrating a precise reference-frame selection rule. This work is a circuit-level quantum information model with no gravity and no dynamical backreaction. It does not claim a black-hole evaporation solution. Instead, it provides an exactly solvable framework showing how information recovery can depend on observable algebra and reference frame in systems with conserved charges. All results are accompanied by reproducibility scripts and verification procedures.
Black-hole hair can affect boundary mixed-state entanglement through local stress-energy and through topological information invisible to the classical metric. We separate these channels for a Nielsen--Olesen vortex on a nonrotating BTZ black hole. A superselection theorem shows that Shannon sector entropy cancels from the Markov gap, while standard U(1)U(1)U(1) symmetry-resolved reflected entropy in a thermal U(1)kU(1)_kU(1)k​ CFT is equipartitioned at leading order, excluding a universal log⁡∣n∣\log|n|log∣n∣ term in the imbalance-resolved gap. We therefore define a Wilson-threaded reflected moment in a probe U(1)kU(1)_kU(1)k​ Chern--Simons completion coupled to the compact vortex-flux class. In a reflected-replica sector with linking number ννν, its normalised phase is 2πκpnν/k2πκpnν/k2πκpnν/k, where κ∈Zκ\in\mathbb Zκ∈Z is the mixed topological coupling; for gcd⁡(κν,k)=1\gcd(κν,k)=1gcd(κν,k)=1, a discrete Fourier transform reconstructs n mod kn\bmod knmodk. The RT connectivity transition switches the specified linked contour on or off, whereas the independent scale ℓ⋆r+/L2=1.128378…\ell_\star r_+/L^2=1.128378\ldotsℓ⋆​r+​/L2=1.128378… determines the direction of the vortex-induced shift of that transition. The charged-moment modulus and the ordinary Markov gap remain geometric observables and are computed from a horizon-anchored Einstein-Abelian-Higgs solution with a complete first-variation kernel including the motion of the entanglement-wedge cross-section endpoints. This phase and modulus separation distinguishes topological hair from gravitational dressing without assigning an unsupported winding-dependent entropy.
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.
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 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.