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arXiv:2607.07342

Horizon-Restricted Leading Soft QED as Open Quantum System

Soo-Jong Rey

hep-thcs.ITgr-qcquant-phSubmitted Jul 8, 2026• Announced Jul 9, 2026• 0 citations
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Abstract

I formulate black-hole-horizon-induced decoherence of charged branch codes as the leading-soft QED restricted to an exterior algebra, formulated as an open quantum system. The fixed-history Feynman--Vernon identity ${\cal F}[J,J]=1$ remains exact. Decoherence enters through the unequal-history influence factor that survives exterior monitoring and belongs to the complementary horizon output. In the coherent eikonal regime, I derive the completely positive Schur channel $({\cal E}_H^{(0)}ρ)_{ab}=\langleΦ_b^{H,(0)}|Φ_a^{H,(0)}\rangle \, ρ_{ab}$. The leading soft input is the eikonal factor, projected onto the horizon radiative algebra. The channel yields Gram-positivity constraints, an exterior quantum-eraser bound, finite-time non-Markovianity tests, soft/hard scaling criteria, and a charged-qutrit interferometer measuring a leading-soft Bargmann holonomy. The holonomy phase is the rephasing-invariant symplectic area of a triangle in horizon soft phase space. I show that its orientation, common-mode, triangulation, and completely positive determinant identities render falsifiable tests beyond pairwise two-path visibility.

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32 pages, 3 figures embedded

Related papers

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I formulate black-hole-horizon-induced decoherence of charged branch codes as the leading-soft QED restricted to an exterior algebra, formulated as an open quantum system. The fixed-history Feynman--Vernon identity F[J,J]=1{\cal F}[J,J]=1F[J,J]=1 remains exact. Decoherence enters through the unequal-history influence factor that survives exterior monitoring and belongs to the complementary horizon output. In the coherent eikonal regime, I derive the completely positive Schur channel (EH(0)ρ)ab=⟨ΦbH,(0)∣ΦaH,(0)⟩ ρab({\cal E}_H^{(0)}ρ)_{ab}=\langleΦ_b^{H,(0)}|Φ_a^{H,(0)}\rangle \, ρ_{ab}(EH(0)​ρ)ab​=⟨ΦbH,(0)​∣ΦaH,(0)​⟩ρab​. The leading soft input is the eikonal factor, projected onto the horizon radiative algebra. The channel yields Gram-positivity constraints, an exterior quantum-eraser bound, finite-time non-Markovianity tests, soft/hard scaling criteria, and a charged-qutrit interferometer measuring a leading-soft Bargmann holonomy. The holonomy phase is the rephasing-invariant symplectic area of a triangle in horizon soft phase space. I show that its orientation, common-mode, triangulation, and completely positive determinant identities render falsifiable tests beyond pairwise two-path visibility.
I formulate unresolved soft-photon sector of QED as open quantum system. Resolved charged particles and hard photons form the system, unresolved soft photons form the environment, and basic object is the reduced density matrix. A resolved outcome fff of multiple hard particles and photons has probability Pf(i)=∑n∣⟨f,n∣S∣i,0⟩∣2=⟨i∣Ff∣i⟩P_{f}(i) =\sum_n|\langle f,n|S|i,0\rangle|^2=\langle i|F_f|i\ranglePf​(i)=∑n​∣⟨f,n∣S∣i,0⟩∣2=⟨i∣Ff​∣i⟩, with Kraus operators Kn=soft⟨n∣S∣0⟩softK_n= {}_{\rm soft} \langle n|S|0 \rangle_{\rm soft}Kn​=soft​⟨n∣S∣0⟩soft​ and effect Ff=∑nKn†ΠfKnF_f=\sum_nK_n^\daggerΠ_fK_nFf​=∑n​Kn†​Πf​Kn​. The SK formulation places unresolved virtual and real terms in one doubled-contour expansion. At one loop they carry the same on-shell eikonal kernel with opposite signs. This elegantly organizes the QED probability: for the same observable, perturbative order, diagrams, and phase space, the OQS gives the same infrared-finite terms as the full-QED. The soft-photon evolution is a unitary coherent-state displacement driven by the scattering current. The equal-history identity of influence functional exactly normalizes this soft evolution; together with the soft-photon theorem it removes the IR divergent leading-soft factor from inclusive probability. I also derive explicit leading-soft QED realization of scale-parametrized Lindblad evolution on a fixed hard-branch space. Tracing an infinitesimal soft-photon shell produces diagonal jump operators whose entries are fixed by the corresponding eikonal emission amplitudes. The finite-shell map is a completely positive unital Schur channel and, in the sharp scale-invariant leading-soft regime, a dephasing semigroup of a completely-positive-divisible scale flow. The resulting logarithmic visibility slope and monotonic purity loss are off-diagonal predictions of the reduced-state description. The same controlled-displacement dilation gives the Sudakov probability, Poisson soft-photon multiplicities, and the bremsstrahlung number spectrum.
The discard ZX-calculus, a diagrammatic language for mixed-state quantum mechanics, is used to give a nonperturbative, categorical proof of the Bloch-Nordsieck cancellation of infrared divergences in QED. Soft photons are treated as an open quantum system: the resolved charged particles and hard photons form the system, while photons below a detector resolution form the environment. The reduced hard channel is a completely positive trace-preserving (CPTP) map, and the soft-photon theorem replaces the full S-matrix by a controlled displacement operator whose Feynman-Vernon influence functional satisfies the equal-history normalization ${\cal F}[J,J]=1 $. In the ZX-calculus, this normalization is a single diagrammatic identity: the doubled displacement diagram collapses to the bare wire under the unitarity, cyclicity, and discard rules. The proof therefore serves as a categorical consistency check on the open-system treatment of soft QED given in a companion paper; it confirms that the physical derivation is logically complete and free of hidden assumptions about the infrared limit. For off-diagonal hard-state elements, the same diagram yields the coherent-state overlap, giving a first-principles account of soft-cloud decoherence. The soft-shell coarse graining is then constructed as a CPTP Schur channel whose infinitesimal limit produces the exact Lindblad generator with jump operators determined by the eikonal emission amplitudes. Finally, a local CPTP-certification pipeline is developed for non-Markovian process tensors, enabling constant-time verification of trace preservation in open quantum simulations. The framework bridges categorical quantum semantics, non-equilibrium field theory, and practical open-system compilation.
Built upon infrared QED formulated as an open quantum system, I use the complementary soft photons as a quantum instrument. The experiment selects the hard-state code, resolved process, trajectory, accepted modes, and photon readout. QED fixes the interaction. On an eikonal-degenerate reducing code, the momentum-flow jump acts as a scalar. A Lindbladian displacement removes this common term from the dissipator. Its interference with the Low--Burnett--Kroll operator enters a finite Hamiltonian. The residual dissipator Jλ(q^){\cal J}_\lambda(\hat q)Jλ​(q^​) measures direction- and helicity-dependent projections of orbital and spin angular momentum. I isolate the same operator in a selective photon measurement by coherently nulling the branch-independent eikonal output. The nulled click effects and the logical dissipator scale as Ed2E_d^2Ed2​. Both saturate at the lower infrared endpoint. In the semiclassical branch limit, the decoherence functionalΓab(1)(Ed)=e22∑λ=±∫0Edd3q(2π)32ω∣Jλ,a(q^)−Jλ,b(q^)∣2 \Gamma^{(1)}_{ab}(E_d)= \frac{e^2}{2}\sum_{\lambda=\pm}\int_0^{E_d} \frac{{\rm d}^3q}{(2\pi)^3 2\omega} \left|{\cal J}_{\lambda,a}(\hat q)-{\cal J}_{\lambda,b}(\hat q)\right|^2Γab(1)​(Ed​)=2e2​∑λ=±​∫0Ed​​(2π)32ωd3q​∣Jλ,a​(q^​)−Jλ,b​(q^​)∣2sets the soft-output overlap. Coherent prescribed currents exponentiate this functional. Repeated outputs give the exact sample complexity for hard-branch identification. General mixed outputs obey the quantum Chernoff bound. I also specialize privacy--correctability duality to the vacuum-plus-one-photon instrument. The resulting criterion identifies hard codes that hide their logical state from the retained soft output.I extend the instrument to orthogonal mode restrictions and zero-mean Gaussian soft states. I prove covariance under unitary mode transformations. I derive the equivalent positive noise-kernel form. The result promotes the finite two-path record to an operator-valued QED instrument. Soft radiation becomes a controlled probe of angular momentum.
It was recently shown that black holes decohere any quantum superpositions in their vicinity. This decoherence is mediated by soft radiation through the horizon, and can be understood as the result of the fact that quantum states in the exterior source distinguishable states of long-range fields in the interior. To study this phenomenon and others, we extend Tomita-Takesaki theory to accommodate states of soft radiation such as arise in the electromagnetic and gravitational memory effects, and provide a general framework for computing the distinguishability of general coherent states. Applying these tools, we use the methods of unambiguous state discrimination and approximate quantum error correction to prove some new relations regarding the distinguishability of quantum states, and the quantum information content of soft radiation, and thereby show that a black hole (or any horizon) decoheres its environment as though its interior were full of optimal observers.
We expand the Einstein--Hilbert action to cubic order about the Schwarzschild horizon, in the even Regge--Wheeler gauge of the Gaddam--Groenenboom--'t~Hooft near-horizon framework, and derive the cubic graviton self-interaction. Our central result is a vanishing theorem: at leading soft order the self-coupling of the purely traceless longitudinal polarizations is identically zero. This is a framework-specific statement: even RW gauge, GGV sector, leading soft order. The surviving interaction lives in the trace sector; its on-shell equal-ℓ\ellℓ weight W(λ,λ,λ)=−3λ(2λ2+λ+3)/(λ+1)2W(λ,λ,λ)=-3λ(2λ^2+λ+3)/(λ+1)^2W(λ,λ,λ)=−3λ(2λ2+λ+3)/(λ+1)2, follows from two independent derivations agreeing to machine precision. We simulate the resulting Hamiltonian at two complementary scales. On IBM Qiskit/Aer, with exact cross-checks, the hardware-format circuits confirm quantitatively what two structural facts already predict, a conserved charge that only the cubic vertex violates (opening φφ→hh→4φφφ\to hh\to4φφφ→hh→4φ) and a resonance-free boost spectrum (gap →1/2\to1/2→1/2): the longitudinal channel is perturbatively rigid, and the simulation measures the residual dressing (deff=1.06d_{\rm eff}=1.06deff​=1.06; multiplicity far from thermal, Poisson, and Haar). A symmetry-graded matrix-product-state evolution then climbs the multipole ladder to sixty modes, the 120120120-qubit register, and settles the one question the multiplet cannot: on a late-time measure common to both engines the inelasticity converges to ηˉ∞≃0.16\barη_\infty\simeq0.16ηˉ​∞​≃0.16 (understated by the single multiplet), the dressing stays area-law with peak entanglement ≃0.45\simeq0.45≃0.45 nats far below the Page value, and the result is cutoff-converged and robust to 969696 qubits at larger occupation. The rigidity therefore persists rather than scrambling as the register grows.
Killing horizons decohere stationary quantum superpositions at a steady rate through the emission of soft radiation (Danielson-Satishchandran-Wald). We ask what information structure this decoherence builds on the horizon. We prove two negative results: (i) the branch-to-branch relative entropy on any exposed local horizon-interval algebra remains bounded in time; (ii) the mutual information between the which-path qubit and any horizon subsystem is bounded by 2 ln 2, so the linear growth is not information acquisition. In the standard coherent-branch model we obtain the exact closed form I(T) = 2 H2((1+exp(-N/2))/2), saturating at 2 ln 2 on the decoherence timescale. What grows linearly is redundancy: disjoint bands of affine frequency generate exactly commuting subalgebras (via a weighted Meyer window construction), and each e-fold of Killing time opens one new soft band re-recording the same which-path bit. We derive the exact fragment formula for I(S:F), exhibit the classical plateau at ln 2, and obtain the linear redundancy law R_delta(T) = N(T) / (2 ln(1/r*(delta))). The record resides in edge (soft-hair) observables, not in local gauge-invariant marginals. Cosmological horizons enforce classical objectivity by Darwinian broadcast, complementing the recent 'optimal interior observer' picture with a many-independent-observers decomposition of the same radiation.