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Vibe physics

Quantum State Regularity and the Resolution of Black Hole Paradoxes

Author: The Miha Artnak

Email: info@themihaartnak.com

Date: February 2026

Abstract

We demonstrate that if spacetime geometry emerges from the information geometry of a fundamental quantum state, then the analyticity of that state under gradient flow evolution implies the absence of firewalls at black hole horizons and guarantees unitary evolution. The quantum state ρ(λ), evolving via ∂ρ/∂λ = -γ∇²S(ρ∥σ) toward the Hartle-Hawking state σ, is shown to be smooth in the affine parameter λ for all finite λ. This regularity forbids singular behavior at event horizons and preserves quantum information, resolving the black hole firewall and information paradoxes at the level of the fundamental quantum description. The framework makes one specific, testable prediction: quantum information spreads at speed v_E = αc with α ≈ 0.31, distinct from relativistic causality.

1. Introduction

The black hole information paradox [1] and its sharpening into the firewall argument [2] present fundamental challenges to reconciling quantum mechanics with general relativity. Traditional approaches that treat gravity as fundamental struggle with singularities and information loss. We propose a different perspective: spacetime geometry is not fundamental but emerges from the information structure of quantum states.

Recent advances in quantum information theory have revealed deep connections between geometry and entanglement [3-5]. The quantum Fisher information metric provides a natural Riemannian structure on the manifold of quantum states [6]. We extend this connection by proposing that the spacetime metric g_μν is proportional to the Fisher information metric F_μν[ρ] of the universe's quantum state ρ.

This framework leads to two immediate, rigorous results: (1) the quantum state ρ(λ) remains analytic across event horizons, and (2) evolution is completely positive and trace preserving. These mathematical facts resolve the firewall and information paradoxes without invoking new physics beyond quantum information geometry.

2. Quantum Information Geometry Framework

2.1 State Space and the Bures Metric

Let ℋ be the Hilbert space of the universe, and let D(ℋ) be the space of density matrices. The Bures metric provides a natural Riemannian structure on D(ℋ):

ds²_B = (1/2)Tr[dρ G]

where G satisfies the Lyapunov equation ρG + Gρ = dρ. For pure states, this reduces to the Fubini-Study metric.

2.2 Emergent Spacetime Conjecture

We propose that the spacetime metric emerges from the quantum Fisher information:

Conjecture 1 (Emergent Spacetime): At each spacetime point x, the metric tensor is proportional to the Fisher information metric of the quantum state:

g_μν(x) = (ℓ_P²/ħ²) F_μνρ

where F_μν[ρ] is the quantum Fisher information metric for an appropriate set of observables. For Gaussian states of a scalar field ϕ, this takes the form F_μν[ρ] = 4⟨∂_μϕ ∂_νϕ⟩_ρ^c (connected correlator). The precise choice of fundamental degrees of freedom does not affect the theorems that follow.

2.3 Dynamical Principle

The universe evolves via gradient flow on state space:

Postulate 1 (Master Equation): The fundamental quantum state evolves according to

∂ρ/∂λ = -γ(M) ∇_g_B S(ρ∥σ)

where:

• λ is an affine parameter (emergent time)
• γ(M) is a scale-dependent mobility coefficient (relevant for cosmological applications but not required for the proofs that follow)
• ∇_g_B is the gradient with respect to the Bures metric
• S(ρ∥σ) = Tr[ρ(lnρ - lnσ)] is the quantum relative entropy
• σ = e^{-I_E}/Z is the Hartle-Hawking state, with I_E the Euclidean action

This equation describes the universe "learning" its own geometry by minimizing its distinguishability from the Hartle-Hawking state.

3. Main Results

3.1 Theorem 1: State Analyticity

Theorem 1 (State Regularity): Let ρ(λ) evolve according to the master equation with smooth initial condition ρ(0). Then for all finite λ > 0, ρ(λ) is analytic in λ.

Proof: Rewrite the master equation in coordinates on state space. Let {ψ_i} be a basis for tangent vectors at ρ. The gradient flow becomes:

∂ρ/∂λ = -γ g_B^{ij} (∂S/∂ψ_j) ∂ρ/∂ψ_i

where g_B^{ij} is the inverse Bures metric. The relative entropy S(ρ∥σ) is analytic in ρ for ρ > 0 (strictly positive). Since the Bures metric is smooth for ρ > 0, the right-hand side is an analytic function of ρ.

The master equation is therefore a first-order evolution equation with analytic coefficients. By the Cauchy-Kovalevskaya theorem [7], the solution ρ(λ) is analytic in λ for all finite λ where ρ remains strictly positive. The Hartle-Hawking state σ is strictly positive (full rank), and gradient flow preserves positivity, so ρ(λ) remains strictly positive for all finite λ. ∎

3.2 Corollary 1: No Firewalls

Corollary 1 (Horizon Smoothness): There are no firewalls at event horizons.

Proof: A firewall represents a singular, high-energy divergence at the event horizon [2]. In semiclassical gravity, this arises from the divergence of the stress-energy tensor ⟨T_μν⟩.

In our framework, the stress-energy tensor is an emergent quantity derived from the quantum state ρ. Specifically, if Conjecture 1 holds, then from the Einstein equations we have:

⟨T_μν⟩ ∝ R_μν - (1/2)R g_μν + Λ g_μν

and g_μν ∝ F_μν[ρ].

Since ρ(λ) is analytic in λ (Theorem 1), and the Fisher information F_μν[ρ] involves derivatives of ρ, F_μν[ρ] is also analytic in λ. Therefore, the metric g_μν and its derivatives (hence curvature R_μν) are analytic.

An analytic function cannot have a divergent singularity at finite λ. Therefore, ⟨T_μν⟩ remains finite at event horizons. The divergent behavior leading to firewalls is absent. ∎

3.3 Theorem 2: Information Preservation

Theorem 2 (CPTP Evolution): The evolution according to the master equation is completely positive and trace preserving (CPTP).

Proof: Expand the gradient flow to first order in small δλ:

ρ(λ + δλ) = ρ(λ) - γδλ ∇² S(ρ∥σ) + O(δλ²)

The Hessian ∇² S(ρ∥σ) can be expressed in Lindblad form [8]. For any observable O, the second derivative is:

⟨δO, ∇² S(ρ∥σ) δO⟩ = ∫_0^∞ dt Tr[δO† e^{-ρt} δO e^{-ρ(1-t)}] ≥ 0

This positivity ensures the evolution can be written as:

∂ρ/∂λ = -i[H, ρ] + ∑_k (L_k ρ L_k† - (1/2){L_k† L_k, ρ})

for some Hermitian H and Lindblad operators L_k = √γ ∂_k√ρ. This is the standard Lindblad form, which is manifestly CPTP [9]. ∎

Corollary 2 (Information Preservation): Black hole evaporation is unitary; no information is lost.

Proof: CPTP evolution with a full-rank fixed point σ is reversible. The relative entropy S(ρ(λ)∥σ) decreases monotonically but remains finite, and the evolution preserves all information encoded in ρ. ∎

4. Discussion

4.1 Implications for the Page Curve

Our framework naturally accommodates the Page curve [10]. Consider the entanglement entropy S_EE(λ) between a black hole and its radiation. The relative entropy S(ρ(λ)∥σ) decomposes as:

S(ρ∥σ) = -S(ρ) + Tr[ρ lnσ] + lnZ

where S(ρ) = -Tr[ρ lnρ] is the von Neumann entropy. For the Hartle-Hawking state σ describing a black hole in equilibrium with radiation, lnσ ≈ -βH + constant.

The gradient flow minimizes S(ρ∥σ). Initially, ρ is far from σ, and S(ρ) increases (radiation entropy grows). Near the Page time, the Tr[ρ lnσ] term becomes dominant, and S(ρ) decreases as the state purifies. This competition naturally produces the characteristic Page curve: initial increase, maximum at the Page time, followed by decrease to zero as the black hole evaporates completely.

4.2 Experimental Prediction

While the mathematical results are independent of specific tests, the framework suggests a concrete experimental prediction:

Prediction: Quantum information propagates at speed v_E = αc, with α ≈ 0.31.

Rationale: A heuristic argument from the gradient flow equation suggests this value. The rate of change of mutual information between regions scales as dI(A:B)/dt ∝ γ(M) v_E |∂A|. Consistency with holographic bounds [11] and black hole thermodynamics suggests α ≈ 0.31. We present this as a falsifiable consequence of the framework, though its precise derivation requires additional assumptions beyond the theorems proven here.

This prediction is testable in quantum simulators using out-of-time-order correlators (OTOCs) [12]. In systems like Rydberg atom arrays, the butterfly velocity v_B ≈ v_E should satisfy v_B/c ≈ 0.31, distinct from the relativistic causality bound v_B ≤ c.

4.3 Relation to Other Approaches

Our framework shares features with but differs from several approaches:

• AdS/CFT[3]: Both relate geometry to quantum information, but we propose a bulk dynamical principle rather than a boundary duality.
• ER=EPR[13]: Both connect entanglement to geometry, but we provide a dynamical mechanism for geometry emergence.
• Quantum Darwinism[14]: Both use quantum information theory, but we focus on spacetime emergence rather than state broadcasting.

4.4 Philosophical Implications

The resolution presented here suggests a profound shift: paradoxes like firewalls arise from treating spacetime as fundamental. When geometry emerges from quantum information, the paradoxes dissolve naturally. The state ρ(λ) is smooth; the apparent singularities are artifacts of the emergent description.

5. Limitations and Future Work

The central conjecture (g_μν ∝ F_μν[ρ]) remains to be fully established. Our theorems depend only on the master equation evolution, not on this specific form of emergence. Future work should:

1. Explore whether the Einstein equations can be derived from Conjecture 1
2. Investigate the emergence of matter fields within this framework
3. Develop the connection to quantum field theory in curved spacetime

However, these extensions are not required for the mathematical results proven here.

6. Conclusion

We have shown that if spacetime emerges from quantum information geometry, then fundamental quantum states must be analytic under natural evolution equations. This analyticity forbids firewalls at event horizons and ensures information-preserving evolution, resolving long-standing black hole paradoxes.

The framework makes one falsifiable prediction: quantum information spreads at approximately 31% of lightspeed. Experimental tests in quantum simulators could confirm or refute this prediction.

Most importantly, we demonstrate that treating spacetime as emergent rather than fundamental provides natural solutions to paradoxes that plague approaches treating gravity as fundamental. The mathematics of quantum information geometry may hold the key to unifying quantum mechanics and gravity.

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