Curvature-Induced Decoherence and Information Recovery

1. Introduction

The interplay between quantum mechanics and curved spacetime appears most pronounced in black-hole evaporation, where semiclassical arguments suggest that an initially pure state evolves into a mixed thermal state [1]. This tension between quantum unitarity and semiclassical gravity underlies the black-hole information paradox [7,8]. Two distinct perspectives have developed:

1.

Decoherence and open-system approaches. Metric fluctuations, tidal forces, and short-wavelength geometric modes have long been argued to induce effective decoherence in quantum fields [9,10,12,13,14]. In stochastic gravity, such effects arise from noise kernels built from graviton two-point functions.

2.

Nonperturbative gravitational saddles. Replica-wormhole contributions to the gravitational path integral show that the fine-grained entropy of Hawking radiation follows the unitary Page curve [19,20,23].

What remains lacking is a single covariant framework that relates the observer-dependent decoherence seen in semiclassical gravity to the unitarity-restoring correlations captured by replica wormholes. Existing decoherence models are typically noncovariant, gauge-dependent, or rely on linearized perturbations of the metric, while replica-wormhole calculations do not describe how semiclassical observers experience progressive loss of purity during evaporation.

1.1. Limitations of Previous Work

• Most gravitational decoherence models assume specific gauges or weak-field expansions, and do not express the decoherence rate in terms of geometric invariants.
• Open-system treatments in curved spacetime usually rely on particular stochastic noise kernels; few provide closed-form bounds on purity evolution.
• No previous work has connected curvature-induced decoherence to an entropy functional with a provable monotonicity property resembling Perelman’s $\mathcal{F}$-entropy.
• The relation between semiclassical decoherence and replica-wormhole unitarity recovery has remained conceptual rather than quantitative.

1.2. Motivating Example: Near-Horizon Curvature and Decoherence

A simple illustration shows the type of behavior we wish to capture. For a Schwarzschild black hole of mass M, the Kretschmann scalar grows as

$$K=R_{\mu \nu \rho \sigma }{R}^{\mu \nu \rho \sigma }={\displaystyle \frac{48M^2}{r^6}},$$

which increases rapidly near the horizon. Any decoherence rate of the form $\Gamma \left(x\right)\propto \sqrt{K\left(x\right)}$ therefore grows as $r^{-3}$, suggesting that semiclassical observers naturally see increased mixedness as wavepackets approach the horizon. This motivates a curvature-dependent deformation of the wavefunctional and a local, covariant open-system evolution.

1.3. Goals and Scope of This Work

The aim of this paper is modest but well defined: We provide a covariant, curvature-dependent Lindblad framework for observer-level decoherence and derive rigorous bounds on purity decay, while clarifying precisely how this semiclassical description fits together with replica-wormhole unitarity restoration.

We distinguish sharply between:

• mathematically derived results: purity-decay theorem, curvature-dependent bounds, monotonicity of the geometric entropy functional;
• heuristic proposals: the interpretation of $\Gamma \left(x\right)$ in terms of geometric fluctuations and the analogy to Perelman-type entropy flows.

1.4. Summary of Contributions

• Covariant curvature-dependent decoherence. We define a curvature-dependent deformation of the matter wavefunctional together with local Lindblad operators $L\left(x\right)$ built from QFT on curved spacetime, yielding a fully covariant open-system evolution.
• Purity-decay theorem. We prove a general inequality showing that purity decreases at a rate controlled by the integrated curvature-dependent parameter ${\Gamma }_{\mathrm{eff}}\left(t\right)$, and that finite-time complete mixing occurs precisely when this integral diverges.
• Geometric entropy functional. We introduce a Perelman-inspired information-theoretic functional $S[g,\psi ]$ and derive a monotonicity formula under a nonphysical, information-theoretic flow. This provides geometric control over the spreading of wavefunctionals.
• Bridge to replica wormholes. We clarify how semiclassical curvature-induced decoherence is reconciled with global unitarity: the semiclassical Lindblad picture traces out geometric modes that reappear as wormhole-induced correlations in the full path integral. A JT-gravity example shows this mechanism explicitly.

The resulting framework places semiclassical decoherence, curvature-driven purity loss, and nonperturbative information recovery within a unified and covariant structure. It does not attempt to provide a microscopic theory of quantum gravity, but rather to supply a mathematically controlled bridge between semiclassical coarse-graining and the unitary evolution encoded in the full gravitational path integral.

2. Curvature-Dependent Lindblad Model

In this section we present a unified treatment of (i) the operator-domain assumptions required for later use of curvature-dependent Lindblad operators, and (ii) the physical and geometric derivation of the local decoherence rate $\Gamma \left(x\right)$ arising from metric fluctuations. Our presentation follows standard frameworks in algebraic and microlocal QFT in curved spacetime [31,32].

2.1. Operator Domains and Admissible Lindblad Operators

Local fields $\varphi \left(x\right)$ and $\pi \left(x\right)$ are operator-valued distributions acting on a common dense invariant domain $\mathcal{D}$, typically the finite-particle subspace of the Fock representation. Essential self-adjointness on $\mathcal{D}$ follows from the microlocal spectrum condition and the existence of Hadamard states, which we assume throughout. These assumptions guarantee well-defined stress-tensor expectation values and fluctuations [31].

For any smooth, compactly supported test function $f\in C_0^{\infty }\left(\Sigma \right)$ the smeared operators

$$L\left(f\right)={\int }_{\Sigma }f\left(x\right)L\left(x\right)d\mu$$

are well defined on $\mathcal{D}$, and all domain subtleties are absorbed by working exclusively with smeared expressions. As in standard algebraic treatments [32], this suffices for constructing completely positive evolution equations.

The curvature-dependent GKLS generator introduced later allows local noise operators of the form

$$L\left(x\right)\in \left\{\varphi \right(x),\pi (x),F(K\left(x\right)\left)\varphi \right(x\left)\right\},$$

where $K=R_{\mu \nu \rho \sigma }{R}^{\mu \nu \rho \sigma }$ is the Kretschmann scalar and F is smooth and bounded. Boundedness ensures that $F\left(K\right)\varphi$ acts on the same domain as $\varphi$ and remains covariant. These assumptions are the only domain conditions required for the purity theorem of Section 3.

2.2. Physical Origin of Curvature-Dependent Decoherence

In semiclassical and stochastic gravity, metric perturbations act as an environment for matter fields, producing an effective nonunitary evolution for the reduced density matrix [12,13]. Let

$$g_{\mu \nu }\to g_{\mu \nu }+h_{\mu \nu }$$

with $h_{\mu \nu }$ a Gaussian perturbation described by its noise kernel $N_{\mu \nu \rho \sigma }(x,x^{\prime })=\langle h_{\mu \nu }\left(x\right)h_{\rho \sigma }\left(x^{\prime }\right)\rangle$. To leading order the coupling to a scalar field produces an influence functional whose quadratic part induces decoherence.

In many spacetimes the short-distance scaling of the noise kernel is controlled by curvature invariants. For example, a schematic estimate

$$\langle h\left(x\right)h\left(x^{\prime }\right)\rangle \sim {\displaystyle \frac{1}{K\left(x\right)}}\delta (x,x^{\prime })$$

is obtained from fluctuational solutions of the linearized Einstein equation on curved backgrounds [13]. Tracing over $h_{\mu \nu }$ then generates a local dissipative term of the form

$$\mathcal{L}\left(\rho \right)={\displaystyle \frac{1}{2}}{\int }_{{\Sigma }_t}d{\mu }_t\Gamma \left(x\right)\left(L\left(x\right)\rho L\left(x\right)-{\displaystyle \frac{1}{2}}\{L{\left(x\right)}^2,\rho \}\right),$$

provided $\Gamma \left(x\right)$ is identified with the appropriate contraction of the noise kernel.

2.3. Dimensional Analysis and Geometric Scaling of $\Gamma \left(x\right)$

The Kretschmann scalar has engineering dimension

$$\left[K\right]=L^{-4},\left[\sqrt{K}\right]=L^{-2}.$$

Since a local decoherence rate must carry dimensions of inverse proper time, a natural covariant form is

$$\Gamma \left(x\right)\sim {\ell }_P^{-2}\sqrt{K\left(x\right)}.$$

The appearance of $\sqrt{K}$ is further supported by (i) the scaling of tidal forces, (ii) metric-fluctuation correlators, and (iii) the fact that generic curvature invariants differ only by order-one dimensionless coefficients for high-curvature regimes. Alternative choices include

$$\Gamma \propto K^{\alpha },\alpha \ge {\displaystyle \frac{1}{2}},\mathrm{or}\Gamma \propto \left|C\right|,\left|R\right|,|R_{\mu \nu }{R}^{\mu \nu }{|}^{1/2},$$

all of which preserve the finiteness/divergence structure required for the purity-decay inequality of Section 3. Our results are insensitive to which invariant is chosen.

2.3.0.1. Examples.

• Schwarzschild: $K=48M^2/r^6$, hence

  $$\sqrt{K}={\displaystyle \frac{4\sqrt{3}M}{r^3}}.$$
  This gives large decoherence near the horizon and rapid blow-up near $r\to 0$.
• Spatially flat FRW: $K=12{(\dot{H}+H^2)}^2+12H^4$, so generically

  $$\sqrt{K}=O\left(H^2\right),$$

  consistent with intuition that cosmic expansion acts as a weak decohering bath.

2.4. Derivation of the Lindblad Generator from Metric Fluctuations

We briefly summarize how curvature-dependent Lindblad dynamics arises from integrating out short-wavelength metric fluctuations. A fully expanded derivation, following the influence-functional methods of [12,13], is included in Appendix A.

2.4.0.2. Total action and linearized metric fluctuations.

Consider a decomposition of the spacetime metric into a smooth background $g_{\mu \nu }$ and a fluctuation $h_{\mu \nu }$:

$$g_{\mu \nu }⟶g_{\mu \nu }+h_{\mu \nu }.$$

The total action is

$$S_{\mathrm{tot}}[g+h,\varphi ]=S_{\mathrm{grav}}[g+h]+S_{\mathrm{matter}}[g+h,\varphi ].$$

Linearizing $S_{\mathrm{matter}}$ in $h_{\mu \nu }$ gives the standard coupling

$$S_{\mathrm{int}}=\frac{1}{2}\int d^{4}x{h}_{\mu \nu }\left(x\right)T^{\mu \nu }\left(x\right)$$

with $T^{\mu \nu }$ the matter stress tensor. We assume the fluctuations h form a Gaussian environment with two-point function

$$N_{\mu \nu \rho \sigma }(x,x^{\prime })={\langle h_{\mu \nu }\left(x\right)h_{\rho \sigma }\left(x^{\prime }\right)\rangle }_g,$$

consistent with stochastic gravity treatments [13] and microlocal Hadamard regularity [32].

2.4.0.3. Influence functional and cumulant expansion.

The reduced density matrix $\rho (\varphi ,{\varphi }^{\prime })$ evolves via the influence functional $e^{i{S}_{\mathrm{IF}}[\varphi ,{\varphi }^{\prime }]}$ obtained by integrating out h in the closed-time-path (CTP) formalism:

$$e^{i{S}_{\mathrm{IF}}[\varphi ,{\varphi }^{\prime }]}:=\int \mathcal{D}h{e}^{i\left(S_{\mathrm{grav}}\left[h\right]+S_{\mathrm{int}}[\varphi ,h]-S_{\mathrm{int}}[{\varphi }^{\prime },h]\right)}.$$

Gaussianity ensures the cumulant expansion terminates at second order:

$$S_{\mathrm{IF}}=-{\displaystyle \frac{1}{8}}\int d^{4}x{d}^4{x}^{\prime }\left[T^{\mu \nu }\left(x\right)-T^{\prime \mu \nu }\left(x\right)\right]N_{\mu \nu \rho \sigma }(x,x^{\prime })\left[T^{\rho \sigma }\left(x^{\prime }\right)+T^{\prime \rho \sigma }\left(x^{\prime }\right)\right].$$

The real part renormalizes the unitary evolution; the imaginary part produces decoherence:

$$\mathrm{Im}{S}_{\mathrm{IF}}={\displaystyle \frac{1}{8}}\int d^{4}x{d}^4{x}^{\prime }\Delta T\left(x\right)N(x,x^{\prime })\Delta T\left(x^{\prime }\right),\Delta T=T-T^{\prime }.$$

2.4.0.4. Nonlocal dissipative kernel.

Insertion into the CTP master equation yields a nonlocal dissipator

$$\dot{\rho }\left(t\right)=-{\int }_{{\Sigma }_t}d{\mu }_t\left(x\right){\int }_{{\Sigma }_{t^{\prime }}}d{\mu }_{t^{\prime }}\left(x^{\prime }\right)N(x,x^{\prime })\left[T\left(x\right),[T\left(x^{\prime }\right),\rho \left(t^{\prime }\right)]\right].$$

(1)

This form is fully general: it contains memory, is non-Markovian, and couples different hypersurfaces ${\Sigma }_t$ and ${\Sigma }_{t^{\prime }}$.

2.4.0.5. Markov and locality limits.

To arrive at a local GKLS form, we adopt the standard approximations used in derivations of open-system master equations:

• Short memory (Markov limit): The correlator $N(x,x^{\prime })$ decays rapidly in the proper-time separation $|t-t^{\prime }|$, allowing

  $$N(x,x^{\prime })\to \delta (t-t^{\prime })\tilde{N}(x,x^{\prime }).$$
• Ultralocality in space: Short-wavelength metric modes dominate, so

  $$\tilde{N}(x,x^{\prime })\approx G\left(x\right){\delta }^{\left(3\right)}(x,x^{\prime }),$$

  where $G\left(x\right)$ is a scalar built from contractions of $N_{\mu \nu \rho \sigma }$ with $T^{\mu \nu }{T}^{\rho \sigma }$.
• Hadamard regularity and smearing: The stress tensor and correlators are implicitly smeared with a smooth test function to control UV singularities [32]. This allows traces and integrals to be interchanged.

Under these assumptions, Eq. (1) collapses to a local GKLS form.

2.4.0.6. Identification of the Lindblad operators.

For a scalar field,

$$T\left(x\right)={\partial }_{\mu }\varphi {\partial }_{\nu }\varphi \left(\mathrm{tensor}\mathrm{contraction}\right),$$

and contractions with $N_{\mu \nu \rho \sigma }$ produce an effective local operator that can be written schematically as

$$L\left(x\right)=F\left(K\right(x\left)\right)\varphi \left(x\right),$$

where $F\left(K\right(x\left)\right)$ is fixed by the local curvature dependence of the noise kernel. Complete positivity follows from the Gaussian influence functional and the Markov–ultralocal limit, exactly as in the standard derivation of quantum Brownian motion.

2.4.0.7. Resulting GKLS generator.

We obtain the curvature-dependent Lindblad equation

$${\displaystyle \frac{d\rho }{dt}}=-i[H,\rho ]-{\int }_{{\Sigma }_t}d{\mu }_t\left(x\right)\Gamma \left(x\right)\left(L\left(x\right)\rho L\left(x\right)-\frac{1}{2}\{L{\left(x\right)}^2,\rho \}\right),$$

(2)

with

$$\Gamma \left(x\right)=G\left(x\right)\ge 0.$$

This is the dynamical law used in the Purity Decay Theorem (Section 3). Covariance follows because $K\left(x\right)$ is a scalar and the construction uses only local contractions of tensorial noise kernels.

3. Curvature-Dependent Lindblad Evolution and Purity Decay

We now develop the mathematical structure of curvature-induced decoherence. Starting from the local GKLS generator introduced earlier, we establish a general and fully rigorous purity-decay inequality. No special ansatz for the state, wavefunctional, or Lindblad operators is required. This result provides model-independent control over semiclassical decoherence driven by spacetime curvature.

3.1. Assumptions and Lindblad Generator

Let $(M,g_{\mu \nu })$ be a globally hyperbolic spacetime foliated by Cauchy hypersurfaces ${\Sigma }_t$, and let ${\rho }_t$ denote a density operator acting on the Fock space of a quantum field. We assume the following.

(A1)

(Hermiticity and domains) Each $L\left(x\right)$ is a densely defined Hermitian operator acting on a common invariant domain $\mathcal{D}$, which contains the finite-particle subspace. This includes the choices

$$L\left(x\right)\in \left\{\varphi \right(x),\pi (x),F(K\left(x\right)\left)\varphi \right(x\left)\right\},$$

with K the Kretschmann scalar and F smooth and bounded. These assumptions conform to the standard microlocal/Hadamard framework [31,32].

(A2)

(Square-integrability) For any $\psi \in \mathcal{D}$,

$${\int }_{{\Sigma }_t}{\parallel L\left(x\right)\psi \parallel }^{2}d{\mu }_t<\infty .$$

(A3)

(Finite effective rate) The curvature-dependent decoherence rate satisfies

$${\Gamma }_{\mathrm{eff}}\left(t\right):={\int }_{{\Sigma }_t}\Gamma (x,t)d{\mu }_t<\infty .$$

(A4)

(Variance bound) The local variance

$$V_{\rho }\left(L\left(x\right)\right):=\mathrm{Tr}\left(\rho L{\left(x\right)}^2\right)-\mathrm{Tr}{\left(\rho L\left(x\right)\right)}^2$$

is bounded from below by a (possibly time-dependent) constant:

$$V_{\rho }\left(L\left(x\right)\right)\ge V_{min}\left(t\right)\ge 0.$$

Under these assumptions, the reduced state of a semiclassical observer satisfies the covariant GKLS equation

$${\displaystyle \frac{d{\rho }_t}{dt}}=-i[H,{\rho }_t]-{\int }_{{\Sigma }_t}\Gamma (x,t)\left(L\left(x\right){\rho }_{t}L\left(x\right)-\frac{1}{2}\{L{\left(x\right)}^2,{\rho }_t\}\right)d{\mu }_t\left(x\right).$$

(3)

3.2. Purity and Variance Identity

Define the purity

$$P\left(t\right)=\mathrm{Tr}\left({\rho }_t^2\right).$$

Differentiating,

$${\displaystyle \frac{dP}{dt}}=2\mathrm{Tr}\left({\rho }_t{\dot{\rho }}_t\right),$$

and using (3), the Hamiltonian contribution vanishes by cyclicity of trace. For any Hermitian $L\left(x\right)$, the standard GKLS identity gives

$$\mathrm{Tr}\left[\rho (L\rho L-\frac{1}{2}{L}^2\rho -\frac{1}{2}\rho L^2)\right]=-V_{\rho }\left(L\right),$$

(4)

where $V_{\rho }\left(L\right)$ is the quantum variance. Substituting into the purity derivative produces a completely general formula:

$${\displaystyle \frac{dP}{dt}}=-2{\int }_{{\Sigma }_t}\Gamma (x,t)V_{\rho }\left(L\left(x\right)\right)d{\mu }_t.$$

(5)

3.3. Purity Decay Theorem

Theorem 1 

(Purity Decay Inequality). Under assumptions(A1)–(A4), the purity of the reduced state satisfies

$${\displaystyle \frac{dP}{dt}}\le -2{\Gamma }_{\mathrm{eff}}\left(t\right)V_{min}\left(t\right).$$

(6)

Moreover, if

$${\int }_0^{t_{*}}{\Gamma }_{\mathrm{eff}}\left(t^{\prime }\right)d{t}^{\prime }=\infty ,$$

(7)

then $P\left(t\right)\to 0$ as $t\to t_{*}$, and the state becomes completely mixed in finite time.

Proof. 

Starting from (5) and invoking (A4),

$${\displaystyle \frac{dP}{dt}}\le -2V_{min}\left(t\right){\int }_{{\Sigma }_t}\Gamma (x,t)d{\mu }_t=-2V_{min}\left(t\right){\Gamma }_{\mathrm{eff}}\left(t\right).$$

This yields the inequality (6). Integrating it gives

$$P\left(t\right)\le exp\left(-2{\int }_0^t{\Gamma }_{\mathrm{eff}}\left(t^{\prime }\right)V_{min}\left(t^{\prime }\right)d{t}^{\prime }\right).$$

If (7) holds and $V_{min}\left(t\right)$ remains positive, the exponent diverges as $t\to t_{*}$, forcing $P\left(t\right)\to 0$. □

3.4. Example: Schwarzschild Evaporation and Page-Time Scaling

For a Schwarzschild black hole of mass $M\left(t\right)$, the Kretschmann scalar at the horizon is

$$K={\displaystyle \frac{48}{{\left(2M\right)}^4}},\sqrt{K}={\displaystyle \frac{4\sqrt{3}}{{\left(2M\right)}^2}}.$$

Under the curvature scaling

$${\Gamma }_{\mathrm{eff}}\left(t\right)\propto M{\left(t\right)}^{-2},$$

and the Hawking evaporation law

$$M\left(t\right)={\left(M_0^3-3\alpha t\right)}^{1/3},$$

we obtain

$${\Gamma }_{\mathrm{eff}}\left(t\right)\propto {\left(M_0^3-3\alpha t\right)}^{-2/3},$$

which diverges as $t\to t_{\mathrm{evap}}$. By Theorem 1, the purity satisfies

$$P\left(t\right)\le exp\left[-2V_{min}{\int }_0^t{\Gamma }_{\mathrm{eff}}\left(t^{\prime }\right)d{t}^{\prime }\right],$$

and the integral diverges as $t\to t_{\mathrm{evap}}$. Thus semiclassical observers necessarily see complete mixing before evaporation is complete. Physically, curvature growth accelerates decoherence near the Page time and diverges near the endpoint of evaporation.

This example illustrates the robustness of the purity-decay theorem: finite-time mixing is governed solely by geometric growth of curvature and does not depend on specific model choices for $L\left(x\right)$ or microscopic details of the underlying quantum gravity theory.

4. Entropy, Information Flow, and Information Geometry

In this section we introduce an entropy functional inspired by geometric analysis but adapted to quantum states on curved spacetime. The goal is not to propose a physical Ricci flow for the spacetime metric, nor to derive any modification of Einstein dynamics. Instead, we develop an information-theoretic analogy that captures how curvature influences the spreading of semiclassical matter wavefunctionals and correlates with the curvature-dependent decoherence discussed in Section 3.

To avoid any misinterpretation, we explicitly separate:

• Formal mathematics: the definition and properties of a geometric information functional;
• Analogy: purely formal comparisons with Perelman-type flows;
• Non-claims: we do not propose new metric dynamics.

4.1. Formal Construction: A Geometric Information Entropy Functional

Given a normalized matter wavefunctional $\psi [g,\varphi ]$ on a globally hyperbolic spacetime $(\mathcal{M},g)$, define the probability density $\rho ={\left|\psi \right|}^2$ and the geometric information entropy

$$\mathcal{S}[g,\psi ]:=-{\int }_{\mathcal{M}}\rho ln\rho \sqrt{-g}{d}^{4}x.$$

(8)

This is the natural Shannon-type functional associated with the probability measure $\rho d{\mu }_g$ on $\mathcal{M}$.

Let $f:=-ln\rho$ so that $\rho =e^{-f}$. Using standard variation formulae for weighted manifolds, one may introduce an entirely auxiliary gradient flow for a tensor field $q_{\mu \nu }$:

$${\partial }_t{q}_{\mu \nu }=-2\left(R_{\mu \nu }\left[q\right]+{\nabla }_{\mu }{\nabla }_{\nu }f\right).$$

(9)

Here $q_{\mu \nu }$ is an auxiliary metric, introduced solely to define a formal gradient flow on weighted probability spaces; it is not the physical metric $g_{\mu \nu }$, and no dynamical interpretation is intended.

Under this auxiliary flow one obtains the monotonicity identity

$${\displaystyle \frac{d}{dt}}\mathcal{S}[g,\psi ]={\int }_{\mathcal{M}}{e}^{-f}\left({|\nabla f|}^2+R\left[g\right]\right)\sqrt{-g}{d}^{4}x\ge 0,$$

(10)

which is purely information-geometric: it states that the curvature-weighted Fisher information is nonnegative.

4.2. Analogy and Interpretation (Nonphysical)

The structure of the auxiliary flow (9) resembles Perelman’s modified Ricci flow and his $\mathcal{F}$-functional. This resemblance is purely formal:

• $q_{\mu \nu }$ is an auxiliary tensor, not the spacetime metric;
• the flow is not derived from the Einstein equations;
• the monotonicity (10) concerns the functional $\mathcal{S}$, not physical geometry.

In this sense, (9) functions analogously to a Fisher–Rao or quantum Fisher gradient flow in information geometry rather than a physical evolution law.

From a mathematical perspective, the functional $\mathcal{S}$ may be viewed as a curvature-weighted Fisher information functional, and the monotonicity (10) may be of independent geometric interest beyond the present physical context.

4.3. Interpretation via Fisher Information and Quantum Geometry

The integrand in (10),

$$e^{-f}{\left(\right|\nabla f|}^2+R\left[g\right]),$$

has a natural interpretation:

• ${|\nabla f|}^2$ is the Fisher information density of the probability distribution $\rho =e^{-f}$;
• the curvature term $R\left[g\right]$ acts as a geometric potential that enhances local distinguishability in regions of large curvature;
• taken together, they resemble the curvature-corrected quantum Fisher information density.

Thus the monotonic growth of $\mathcal{S}$ indicates increasing statistical spread or “information delocalization” of the wavefunctional.

4.4. Toy Example: $1+1$-Dimensional JT Gravity

To illustrate the behaviour of $\mathcal{S}[g,\psi ]$, consider Euclidean JT gravity with metric

$$d{s}^2={\displaystyle \frac{1}{z^2}}(d{z}^2+d{t}^2),R=-2,$$

and a Gaussian localized matter density

$$\rho (z,t)={\displaystyle \frac{1}{2\pi {\sigma }^2}}exp\left[-{\displaystyle \frac{{(z-z_0)}^2+t^2}{2{\sigma }^2}}\right].$$

Then $f=-ln\rho$ and ${|\nabla f|}^2=\left({(z-z_0)}^2+t^2\right)/{\sigma }^4$. Inserting into (10) yields

$${\displaystyle \frac{d\mathcal{S}}{dt}}=\int e^{-f}\left({\displaystyle \frac{{(z-z_0)}^2+t^2}{{\sigma }^4}}-2\right){\displaystyle \frac{dzdt}{z^2}}.$$

One finds ${\displaystyle \frac{d\mathcal{S}}{dt}}>0$ whenever ${\sigma }^2<{\sigma }_{★}^2\left(z_0\right)$, i.e. for sufficiently localized wavefunctionals, most notably those near the near-horizon region ($z_0\to 1$). This matches the intuition that strong curvature enhances spreading of matter probability.

4.5. Connection to Curvature-Induced Decoherence

Although the auxiliary flow (9) is formal and nonphysical, the entropy functional (8) provides a useful heuristic comparison with the curvature-induced decoherence studied in Section 3. In particular:

• regions of large curvature enhance ${|\nabla f|}^2$ and hence increase $\mathcal{S}$;
• the same regions yield large curvature-dependent decoherence rates $\Gamma \left(x\right)$ in the Lindblad generator;
• thus the growth of $\mathcal{S}$ provides a qualitative parallel to the purity-decay inequality

  $${\displaystyle \frac{dP}{dt}}\le -2{\Gamma }_{\mathrm{eff}}\left(t\right)V_{min}\left(t\right),$$

  though no direct dynamical identification is made.

4.6. Summary and Disclaimers

The construction of this section is purely geometric and information-theoretic:

• we do not propose a Ricci flow for the physical metric;
• we do not modify classical or semiclassical gravitational dynamics;
• the auxiliary flow (9) is a bookkeeping device for analyzing the monotonicity of $\mathcal{S}$;
• the connection to decoherence is heuristic and based on qualitative parallels.

Nevertheless, the geometric functional $\mathcal{S}[g,\psi ]$ offers a clean, calculable way to describe curvature-enhanced redistribution of information in semiclassical wavefunctionals, complementing the Lindblad-based purity analysis in Section 3.

5. Islands, Replica Wormholes, and Information Recovery

Nonperturbative developments in the gravitational path integral have dramatically altered the understanding of information flow in evaporating black holes. In the semiclassical description, tracing over the interior degrees of freedom forces an exterior observer to assign a mixed density matrix to Hawking radiation, with entropy that increases monotonically. However, replica computations of fine-grained entropy reveal additional gravitational saddles—replica wormholes—that dominate after the Page time and modify this semiclassical picture, yielding the unitary Page curve [19,20,21].

In this section we strengthen the conceptual bridge between semiclassical curvature-induced decoherence and nonperturbative information recovery, and we illustrate the connection through an explicit 2D JT gravity example.

5.1. Semiclassical vs. Nonperturbative Descriptions

A semiclassical observer necessarily coarse-grains over the black-hole interior, metric fluctuations, short-wavelength geometric modes, and any Planck-scale structure near the horizon. This induces curvature-driven decoherence, leading to:

• mixed exterior density matrices,
• monotonic purity decay (Section 3),
• apparent information loss,
• entropy growth consistent with the geometric flow of (Section 4).

By contrast, the full gravitational path integral includes nonperturbative saddles that connect different replica sheets. These wormhole configurations encode correlations that a semiclassical observer has inadvertently traced out. Including such saddles restores global unitarity and produces a fine-grained entropy that follows the Page curve.

$$\mathbf{Semiclassical}:\mathrm{trace}\mathrm{interior}\Rightarrow \mathrm{decoherence},\mathrm{mixed}{\rho }_{\mathrm{rad}}$$

$$\mathbf{Full}\mathbf{quantum}\mathbf{gravity}:\mathrm{replica}\mathrm{wormholes}\mathrm{included}\Rightarrow \mathrm{purity}\mathrm{restored}$$

This contrast highlights that semiclassical mixedness is an observer-dependent, not fundamental, feature.

5.2. Islands and the Quantum Extremal Surface Formula

Replica-wormhole computations lead to the island formula,

$$S_{\mathrm{rad}}\left(t\right)=\underset{I}{min}\left\{{\displaystyle \frac{\mathrm{Area}(\partial I)}{4G_N}}+S_{\mathrm{matter}}(I\cup \mathrm{rad})\right\},$$

(11)

where I is an island region inside the black hole and $\partial I$ its quantum extremal surface (QES).

Before the Page time, the QES lies outside the horizon and the island is absent. After the Page time, the QES jumps to a surface behind the horizon, thereby enlarging the entanglement wedge of the radiation.

5.3. Worked Example: 2D JT Gravity

To make these ideas concrete, consider Jackiw–Teitelboim (JT) gravity coupled to a conformal matter sector. The Euclidean action is

$$I_{\mathrm{JT}}=-{\displaystyle \frac{1}{2}}\int d^{2}x\sqrt{g}\varphi (R+2)-{\varphi }_b\int Kds.$$

The Hawking temperature is fixed by the AdS₂ radius, and black holes evaporate into the external bath.

Fine-grained entropy before the Page time.

For early times, the QES lies outside the horizon and the entropy of radiation equals the CFT entanglement entropy,

$$S_{\mathrm{rad}}^{\mathrm{early}}\left(t\right)\simeq {\displaystyle \frac{c}{6}}log\left[{\displaystyle \frac{\beta }{\pi ϵ}}sinh\left({\displaystyle \frac{2\pi t}{\beta }}\right)\right],$$

which grows monotonically.

Fine-grained entropy after the Page time.

When $t>t_{\mathrm{Page}}$, the island solution dominates:

$$S_{\mathrm{rad}}^{\mathrm{late}}\left(t\right)\simeq {\displaystyle \frac{{\varphi }_{\mathrm{hor}}}{4G_N}}+O\left(1\right),$$

which is time-independent. The radiation entropy decreases to a constant, producing the descending part of the Page curve.

Curvature-dependent $\Gamma \left(x\right)$ in JT gravity.

In JT gravity the “curvature” is constant, $R=-2$, but the relevant invariant for decoherence is the effective tidal scale associated with the dilaton gradient. The Kretschmann-like invariant is

$$K_{\mathrm{eff}}\left(x\right)\sim {(\nabla \varphi )}^2,$$

and thus the curvature-induced decoherence rate behaves as

$$\Gamma \left(x\right)\propto {\ell }_P^{-2}\sqrt{K_{\mathrm{eff}}\left(x\right)}\sim {\ell }_P^{-2}|\nabla \varphi \left(x\right)|.$$

Near the boundary, the dilaton grows linearly and $\Gamma \left(x\right)$ is small; near the horizon, the dilaton approaches its extremal value and $\Gamma \left(x\right)$ becomes large. This structure mirrors the semiclassical growth of entanglement entropy in JT gravity.

Relation between ${\Gamma }_{\mathrm{eff}}\left(t\right)$ and the Page transition.

Using the evaporating-JT analog of Hawking flux, the effective decoherence rate integrates to

$${\Gamma }_{\mathrm{eff}}\left(t\right)\sim {\int }_{\mathrm{ext}.\mathrm{region}}|\nabla \varphi |d\mu \propto log\left[sinh\left({\displaystyle \frac{2\pi t}{\beta }}\right)\right].$$

Thus ${\Gamma }_{\mathrm{eff}}\left(t\right)$ grows monotonically and diverges at the same rate as the semiclassical entropy $S_{\mathrm{rad}}^{\mathrm{early}}\left(t\right)$. The Page time occurs at the moment when

$$S_{\mathrm{rad}}^{\mathrm{early}}\left(t_{\mathrm{Page}}\right)={\displaystyle \frac{{\varphi }_{\mathrm{hor}}}{4G_N}},$$

which coincides with the moment when ${\Gamma }_{\mathrm{eff}}\left(t\right)$ becomes large enough that curvature-induced decoherence is maximal for the semiclassical observer.

In this sense, the Page transition is the moment when the Lindblad description is most strongly affected by gravitational curvature, while the full quantum gravity description replaces the semiclassical coarse-graining by island correlations.

5.4. Connection to Curvature-Induced Decoherence

The curvature-induced decoherence picture developed in this work describes the semiclassical, observer-dependent evolution of the radiation. The Lindblad generator leads to the purity-decay inequality

$${\displaystyle \frac{dP}{dt}}\le -2{\Gamma }_{\mathrm{eff}}\left(t\right)V_{min}\left(t\right),$$

and to monotonic entropy production (Section 4).

Replica wormholes refine this picture by demonstrating that:

• the geometric modes traced out in the Lindblad description still exist in the full theory,
• their correlations are reintroduced through replica wormholes,
• the global state $|{\Psi }_{\mathrm{total}}\rangle$ remains pure, even though ${\rho }_{\mathrm{rad}}$ is mixed semiclassically.

Thus curvature-induced decoherence and the emergence of islands describe different levels of coarse-graining of the same underlying unitary evolution.

5.5. Summary

Replica wormholes and islands act as the nonperturbative completion of the semiclassical decoherence mechanism. In the effective description, curvature drives decoherence and purity loss; in the full description, global purity is restored once island saddles are included. This reconciles:

• Semiclassical information loss: an artifact of tracing out geometric degrees of freedom;
• Nonperturbative information recovery: an exact property of the gravitational path integral.

The JT gravity example illustrates explicitly how semiclassical decoherence rates $\Gamma \left(x\right)$ grow until the Page transition, at which point island contributions dominate and the Page curve turns over.

6. Discussion and Outlook

This work has developed a unified framework in which semiclassical, curvature-induced decoherence provides the observer-dependent origin of mixedness, while nonperturbative gravitational saddles—islands and replica wormholes—restore global unitarity. The purity-decay inequality of Section 3 and the geometric entropy monotonicity of Section 4 clarify how semiclassical descriptions lose track of correlations that re-emerge nonperturbatively in the full gravitational path integral. The explicit JT-gravity example of Section 5 illustrates this interplay in a setting where both curvature-induced decoherence and nonperturbative information recovery can be computed analytically.

6.1. Limitations of the Present Framework

Despite these conceptual advances, several limitations remain:

1.

Phenomenological form of the decoherence rate. The curvature-dependent Lindblad rate $\Gamma \left(x\right)$, although motivated by dimensional analysis and semiclassical considerations, is not derived from a fully microscopic quantum-gravity calculation. Its relation to metric correlators, graviton self-interactions, or edge-mode dynamics remains to be rigorously established.

2.

Restricted class of spacetimes. Our theorems most naturally apply to quasi-static or slowly evolving geometries admitting global foliations. Strongly dynamical black-hole mergers, rotating spacetimes (Kerr), and Vaidya-like evaporating metrics require additional geometric control over the operator domains and the time-dependence of ${\Gamma }_{\mathrm{eff}}\left(t\right)$.

3.

Semiclassical coarse-graining. The Lindblad description presupposes a separation between “accessible” matter degrees of freedom and “inaccessible” geometric modes. A more precise justification should ideally be phrased in the language of algebraic QFT, modular inclusions, or split-property constructions.

4.

Replica wormholes and operator algebras. Our connection between curvature-induced decoherence and islands is conceptual rather than algebraic. A sharper statement would relate the breakdown of purity to the transition between von Neumann algebras describing exterior radiation before and after the Page time.

These limitations point toward directions in which our framework should be extended.

6.2. Future Directions

Several promising research avenues emerge naturally:

• Microscopic derivation of the decoherence rate. A key open problem is computing $\Gamma \left(x\right)$ from a fundamental theory: loop quantum gravity (spin-network fluctuations), string theory (worldsheet moduli, D-brane backreaction), or the AdS/CFT correspondence (stress-tensor correlators in strongly coupled CFTs). Establishing a universal scaling law would significantly strengthen the physical basis of our results.
• Extension to rotating and dynamical horizons. Curvature scalars such as K grow differently in Kerr and in rapidly evaporating Vaidya geometries. Applying the purity-decay inequality in these settings would test the robustness of curvature-induced decoherence under shear, frame dragging, and time-dependent outer trapped surfaces.
• Nonperturbative constraints from modular theory. Connections with modular Hamiltonians, Tomita–Takesaki theory, and algebraic QFT may clarify how semiclassical coarse-graining corresponds to restrictions on the operator algebra accessible to an exterior observer. Such techniques would also help formalize the island transition as a change in the relevant entanglement wedge algebra.
• Numerical simulations of evaporating geometries. Combining semiclassical evolution equations for the metric with the Lindblad purity equation

  $${\displaystyle \frac{dP}{dt}}\le -2{\Gamma }_{\mathrm{eff}}\left(t\right)V_{min}\left(t\right)$$

  would enable quantitative tests of curvature-induced decoherence in toy models of black-hole evaporation. Such simulations may identify the regimes where the semiclassical picture becomes maximally inaccurate and nonperturbative saddles necessarily dominate.
• Bridging to holography and quantum error correction. The reappearance of interior degrees of freedom in the nonperturbative path integral suggests deep links with quantum error-correcting structures in AdS/CFT. Understanding how curvature-induced decoherence acts on code subspaces could clarify how emergent spacetime encodes coarse-grained versus fine-grained information.

6.3. Outlook

Taken together, these directions may lead to a sharpened and more complete picture of semiclassical information loss: not an indication of fundamental non-unitarity, but an inevitable consequence of tracing over geometric degrees of freedom that re-emerge as nonperturbative correlations in the full gravitational path integral. A microscopic derivation of $\Gamma \left(x\right)$, and a more algebraic understanding of the island transition, would significantly advance the program of reconciling semiclassical decoherence with exact quantum-gravitational unitarity.

Appendix A.1. Setup of the Total System

Let the spacetime metric be decomposed as

$$g_{\mu \nu }\to g_{\mu \nu }+h_{\mu \nu },$$

(A12)

where $g_{\mu \nu }$ is a smooth background and $h_{\mu \nu }$ contains short-wavelength fluctuations. The total action is

$$S_{\mathrm{tot}}[g+h,\varphi ]=S_{\mathrm{grav}}[g+h]+S_{\mathrm{matter}}[g+h,\varphi ].$$

(A13)

We assume that:

1.

$h_{\mu \nu }$ is treated perturbatively to second order;

2.

its statistics are Gaussian with two-point function

$$N_{\mu \nu \rho \sigma }(x,x^{\prime })=\langle h_{\mu \nu }\left(x\right)h_{\rho \sigma }\left(x^{\prime }\right)\rangle ,$$

(A14)

3.

the background state of $h_{\mu \nu }$ obeys the Hadamard condition;

4.

the matter coupling is linearized,

$$S_{\mathrm{matter}}[g+h,\varphi ]=S_{\mathrm{matter}}[g,\varphi ]+{\displaystyle \frac{1}{2}}\int h^{\mu \nu }\left(x\right)T_{\mu \nu }\left(x\right)d^{4}x+O\left(h^2\right).$$

(A15)

The reduced density matrix of matter fields is obtained by tracing out the metric fluctuations:

$${\rho }_{\mathrm{m}}(\varphi ,{\varphi }^{\prime })=\int \mathcal{D}h{e}^{i{S}_{\mathrm{tot}}[g+h,\varphi ]-i{S}_{\mathrm{tot}}[g+h,{\varphi }^{\prime }]}{\rho }_{\mathrm{grav}}\left(h\right).$$

Appendix A.2. Influence Functional and Cumulant Expansion

The reduced dynamics is encoded in the Feynman–Vernon influence functional:

$$\mathcal{F}[\varphi ,{\varphi }^{\prime }]=exp\left(i{S}_{\mathrm{IF}}[\varphi ,{\varphi }^{\prime }]\right).$$

(A16)

Using the linearized coupling (A4) and Gaussianity of h, the cumulant expansion truncates at second order:

$$\begin{array}{cc}\hfill S_{\mathrm{IF}}& ={\displaystyle \frac{i}{8}}\int d^{4}x{d}^4{x}^{\prime }\Delta T_{\mu \nu }\left(x\right)N^{\mu \nu \rho \sigma }(x,x^{\prime })\Delta T_{\rho \sigma }\left(x^{\prime }\right),\hfill \end{array}$$

(A17)

where $\Delta T_{\mu \nu }=T_{\mu \nu }\left[\varphi \right]-T_{\mu \nu }\left[{\varphi }^{\prime }\right]$. The imaginary part of $S_{\mathrm{IF}}$ generates decoherence.

Appendix A.3. Noise Kernel Contraction and Matter Operators

We express the stress tensor as a bilinear in $\varphi$, schematically

$$T_{\mu \nu }\left(x\right)=D_{\mu \nu }[\varphi \left(x\right),\nabla \varphi \left(x\right)],$$

and expand $\Delta T_{\mu \nu }$ in functional derivatives. The key contraction in (A6) produces an effective quadratic form in local matter operators:

$$\mathrm{Im}{S}_{\mathrm{IF}}={\displaystyle \frac{1}{2}}\int d^{4}x{d}^4{x}^{\prime }K(x,x^{\prime })O\left(x\right)O\left(x^{\prime }\right),$$

(A18)

where $O\left(x\right)$ is a local Hermitian matter operator (e.g. $\varphi$, $\pi$, or $F\left(K\right)\varphi$) and

$$K(x,x^{\prime })={\Delta }_{\mu \nu }\left(x\right)N^{\mu \nu \rho \sigma }(x,x^{\prime }){\Delta }_{\rho \sigma }\left(x^{\prime }\right)$$

(A19)

is the induced noise kernel felt by matter.

Appendix A.4. Markov Approximation and Local Limit

The nonlocal kernel $K(x,x^{\prime })$ encodes memory effects. Assuming:

• short correlation time of $N(x,x^{\prime })$ (UV-dominated fluctuations),
• separation of timescales between matter and metric correlation times,
• appropriate smearing on Cauchy slices to ensure convergence,

one may adopt the Markov approximation:

$$K(x,x^{\prime })\approx 2\Gamma \left(x\right)\delta (t-t^{\prime }){\delta }^{\left(3\right)}(\mathbf{x}-{\mathbf{x}}^{\prime }),$$

(A20)

yielding a local dissipation term.

The rate is thus:

$$\Gamma \left(x\right)={\displaystyle \frac{1}{2}}\int d\tau {\Delta }_{\mu \nu }\left(x\right)N^{\mu \nu \rho \sigma }(x,x;\tau ){\Delta }_{\rho \sigma }\left(x\right),$$

(A21)

which depends on local curvature through the graviton two-point function. For Hadamard states, the short-distance expansion of N contains curvature scalars such as $K=R_{\mu \nu \rho \sigma }{R}^{\mu \nu \rho \sigma }$, yielding

$$\Gamma \left(x\right)\sim {\ell }_P^{-2}\sqrt{K\left(x\right)},$$

(A22)

to leading order in derivative expansion.

Appendix A.5. GKLS Form of the Reduced Dynamics

Expanding the influence functional to second order,

$$\rho \to \rho +\delta \rho =\rho +\int dt{\int }_{{\Sigma }_t}d{\mu }_t\Gamma \left(x\right)\left(O\left(x\right)\rho O\left(x\right)-\frac{1}{2}\{O{\left(x\right)}^2,\rho \}\right),$$

one identifies the GKLS generator:

$$\begin{array}{|c|}\hline {\displaystyle \frac{d\rho }{dt}}=-i[H,\rho ]+{\int }_{{\Sigma }_t}d{\mu }_t\Gamma \left(x\right)\left(L\left(x\right)\rho L\left(x\right)-\frac{1}{2}\{L{\left(x\right)}^2,\rho \}\right),\\ \hline\end{array}$$

(A23)

with

$$L\left(x\right)=O\left(x\right),\Gamma \left(x\right)\mathrm{given}\mathrm{by}\left(\right).$$

(A24)

Complete positivity.

Because $h_{\mu \nu }$ is Gaussian and the cumulant expansion truncates at second order, the influence functional is of Kossakowski–Lindblad type. Hence the resulting generator (A12) is automatically completely positive and trace preserving under the Markov assumptions listed above.

Appendix A.6. Assumptions and Validity

The derivation relies on:

• Gaussianity of metric fluctuations;
• Hadamard short-distance structure of $N_{\mu \nu \rho \sigma }$;
• linearized matter–gravity coupling (A4);
• separation of timescales enabling the Markov limit;
• smearing on Cauchy slices ensuring integrability and domain control;
• restriction to operators sharing a common invariant domain.

Within these assumptions, the curvature-dependent GKLS equation (A12) provides a consistent and covariant open-system description of matter fields interacting with metric fluctuations.

Appendix B.1. Setting and Assumptions

We work in a globally hyperbolic background $(M,g)$ and consider linearized metric perturbations $h_{\mu \nu }$ quantized on $(M,g)$ around a Hadamard state. The noise kernel is defined by the two-point function

$$N_{\mu \nu \rho \sigma }(x,x^{\prime })=\langle h_{\mu \nu }\left(x\right)h_{\rho \sigma }\left(x^{\prime }\right)\rangle ,$$

viewed as a bi-distribution on $M\times M$. Our derivation uses the following assumptions:

(B1)

Hadamard short-distance structure. The graviton two-point function has the standard Hadamard singular part; modulo state-dependent smooth terms, the singularity structure is fixed by the local geometry [31,32].

(B2)

UV dominance in high-curvature regions. In regions where curvature invariants are large (in Planck units) short-wavelength modes dominate the local noise kernel; their contribution can be captured by a local expansion in curvature invariants.

(B3)

Gaussian environment approximation. Metric fluctuations are approximated as Gaussian (valid for weakly nonlinear graviton dynamics or when only two-point correlations are retained).

(B4)

Sufficient smearing. All operator-valued distributions are tested against smooth compactly supported functions; this controls UV divergences and permits exchanging integrals and traces.

These are the same assumptions used in stochastic gravity and in controlled influence-functional derivations (Appendix A).

Appendix B.2. Local Curvature Expansion of the Noise Kernel

The Hadamard short-distance expansion implies that for $x^{\prime }\to x$ the noise kernel admits an asymptotic expansion of the schematic form

$$N_{\mu \nu \rho \sigma }(x,x^{\prime })=H_{\mu \nu \rho \sigma }(x,x^{\prime })+W_{\mu \nu \rho \sigma }(x,x^{\prime }),$$

where H contains universal singular terms (state independent) and W is a smooth state-dependent remainder. The singular Hadamard part can be written in Riemann normal coordinates as a series in the Synge world function and local curvature tensors. Schematically,

$$H(x,x^{\prime })\sim {\displaystyle \frac{U(x,x^{\prime })}{\sigma (x,x^{\prime })}}+V(x,x^{\prime })ln\sigma (x,x^{\prime })+\dots ,$$

with $U,V$ smooth bi-tensors constructed from curvature and its derivatives [3,43].

Under the assumption (B2) that UV/short-wavelength modes dominate in regions of large curvature, the smooth part W contains terms whose leading local dependence can be expressed as scalar contractions of curvature. Contracting with appropriate tensor projectors (coming from the stress tensor structure) gives an induced scalar noise kernel $K(x,x^{\prime })$ entering the matter influence functional (see Appendix A). To leading local order one may therefore write, schematically,

$$K(x,x^{\prime })\approx {\displaystyle \frac{a_0}{{\ell }_P^2}}{F}_0\left({\ell }_{\mathrm{curv}}^{-2}\left(x\right)\right)\mathcal{K}(\Delta t,\Delta \mathbf{x})+\mathcal{R}(x,x^{\prime }),$$

(A25)

where:

• $a_0$ is a dimensionless constant encoding contraction coefficients,
• ${\ell }_P$ is the Planck length (we restore factors of ${\ell }_P$ to make scalings explicit),
• ${\ell }_{\mathrm{curv}}\left(x\right)$ is the local curvature length scale defined by typical curvature invariant(s), e.g. ${\ell }_{\mathrm{curv}}^{-4}\sim K\left(x\right)$,
• $F_0$ is a dimensionless function of the dimensionless combination ${\ell }_{\mathrm{curv}}^{-2}{\ell }_P^2$ that reduces to a power-law for high curvature,
• $\mathcal{K}(\Delta t,\Delta \mathbf{x})$ is a short-range correlation kernel that decays on time and spatial scales of order ${\tau }_c\sim {\ell }_{\mathrm{curv}}$,
• $\mathcal{R}(x,x^{\prime })$ is a remainder subleading for $x^{\prime }\to x$.

Appendix B.3. Correlation Time and Markov Limit

From the local structure of (A14) we estimate the correlation time ${\tau }_c\left(x\right)$ of the gravitational environment to be of order the local curvature length,

$${\tau }_c\left(x\right)\sim {\ell }_{\mathrm{curv}}\left(x\right)\sim K{\left(x\right)}^{-1/4}.$$

The Markov (short-memory) approximation used to obtain a local GKLS form is controlled when the typical matter timescale ${\tau }_{\mathrm{m}}$ satisfies ${\tau }_{\mathrm{m}}\gg {\tau }_c$. In high-curvature regions ${\tau }_c$ becomes small and the Markov limit is better justified.

Appendix B.4. Estimate for the Local Decoherence Rate Γ(x)

The local decoherence rate arises from the time-integral of $K(x,x^{\prime })$ contracted with matter two-point functions. A conservative, dimensionally consistent estimate is

$$\Gamma \left(x\right)\sim {\ell }_P^{-2}F\left({\ell }_P^{2}K\left(x\right)\right),$$

(A26)

where F is a dimensionless function determined by the detailed tensorial contractions and the matter correlator. In the high-curvature (short-length) regime we may expand F as a power law:

$$F\left({\ell }_P^{2}K\right)\sim C{\left({\ell }_P^{2}K\right)}^{\alpha },\alpha \ge \frac{1}{2},$$

where $\alpha =\frac{1}{2}$ corresponds to the minimal scaling argued from dimensional analysis in the main text. Thus one finds the parametric bound

$$c_1{\ell }_P^{-2}K{\left(x\right)}^{1/2}\lesssim \Gamma \left(x\right)\lesssim c_2{\ell }_P^{-2}K{\left(x\right)}^{\beta },$$

(A27)

for some $\beta \ge 1/2$ and positive constants $c_1,c_2$ that depend on the state and the matter content.

Remarks on rigor.

The estimate (A15) is a controlled dimensional/short-distance result rather than a strict theorem; its validity rests on assumptions (B1)–(B3) and on smearing the operators. A fully rigorous bound could be produced under stronger spectral assumptions on the graviton two-point function (for example, a detailed microlocal analysis), but for the purposes of the purity theorem the inequalities (A16) suffice: any exponent $\alpha \ge 1/2$ produces the same finite/divergent dichotomy for the integrated rate ${\Gamma }_{\mathrm{eff}}$.

Appendix B.5. Conservative Lower and Upper Bounds

Using the contraction structure and the Hadamard short-distance expansion one obtains the following schematic bounds valid after smearing over test functions $f\in C_0^{\infty }\left({\Sigma }_t\right)$:

$$\begin{array}{cc}\hfill {\int }_{{\Sigma }_t}\Gamma \left(x\right){\left|f\left(x\right)\right|}^{2}d{\mu }_t\left(x\right)& \ge c_1{\ell }_P^{-2}{\int }_{{\Sigma }_t}K{\left(x\right)}^{1/2}{\left|f\left(x\right)\right|}^{2}d{\mu }_t\left(x\right),\hfill \end{array}$$

(A28)

$$\begin{array}{cc}\hfill {\int }_{{\Sigma }_t}\Gamma \left(x\right){\left|f\left(x\right)\right|}^{2}d{\mu }_t\left(x\right)& \le c_2{\ell }_P^{-2}{\int }_{{\Sigma }_t}K{\left(x\right)}^{\beta }{\left|f\left(x\right)\right|}^{2}d{\mu }_t\left(x\right).\hfill \end{array}$$

(A29)

These inequalities are the rigorous form of (A16) when tested against smooth smearing functions. They are sufficient to establish the qualitative statements in the main text about divergence of ${\Gamma }_{\mathrm{eff}}$ when curvature invariants blow up.

Appendix B.6. Summary

Under the Hadamard and short-memory assumptions, local contractions of the metric two-point function produce a decoherence rate with leading scaling $\Gamma \left(x\right)\sim {\ell }_P^{-2}K{\left(x\right)}^{\alpha }$ for some $\alpha \ge 1/2$. The Markov limit is controlled by the local curvature length ${\ell }_{\mathrm{curv}}$ and improves in regions of large curvature. The smeared bounds (A17)–() provide the technical link used in the purity theorem.

Appendix C.1. Model Assumptions

We work in two-dimensional Jackiw–Teitelboim (JT) gravity minimally coupled to a matter field. The classical background metric of a static black hole in (near-)Schwarzschild coordinates may be written as

$$d{s}^2=-f(r;M)d{t}^2+f{(r;M)}^{-1}d{r}^2,f(r;M)={\displaystyle \frac{r^2-r_h{\left(M\right)}^2}{{\ell }^2}},$$

where M parameterizes the ADM mass and $r_h\left(M\right)$ is the horizon radius; the AdS₂ length ℓ sets the curvature scale. In true JT gravity the scalar curvature is constant, $R=-2/{\ell }^2$, but dynamical aspects (the dilaton) and coupling to a thermal bath produce a time-dependence of the mass parameter $M\left(t\right)$. We adopt the following simplified, physically motivated assumptions:

(C1)

The local decoherence rate per unit proper volume has a curvature-driven parametric form

$$\Gamma (r,t)={\Gamma }_0\mathcal{G}(r;M\left(t\right)),$$

where ${\Gamma }_0\sim {\ell }_P^{-2}$ sets the Planckian prefactor and $\mathcal{G}$ encodes the radial dependence. In JT-like spacetimes $\mathcal{G}$ will be localized near the horizon due to blueshift effects.

(C2)

The mass evolves according to a simple evaporation law

$${\displaystyle \frac{dM}{dt}}=-\mu M^{-\gamma },\mu >0,\gamma \ge 0,$$

which captures a family of evaporation regimes. (For 4D Hawking evaporation $\gamma =2$; for 2D models $\gamma$ may differ. We keep $\gamma$ general.)

(C3)

The spatial slice ${\Sigma }_t$ is chosen to extend from $r_h\left(M\left(t\right)\right)$ to a fixed cutoff $r_{\max }$ representing the asymptotic bath region. The proper-volume element on ${\Sigma }_t$ is $d{\mu }_t=\sqrt{g_{rr}}dr$.

Appendix C.2. A Concrete Choice for G

A simple model that captures horizon-localized decoherence is

$$\mathcal{G}(r;M)={\left({\displaystyle \frac{r_h\left(M\right)}{r}}\right)}^p,p>0,$$

so that decoherence is strongest near the horizon $r\approx r_h$ and decays away from it. This is a toy choice consistent with the idea that curvature intensity (or tidal blueshift) scales like an inverse power of r near the horizon.

Appendix C.3. Computation of Γ_eff (t)

The integrated rate is

$${\Gamma }_{\mathrm{eff}}\left(t\right)={\int }_{{\Sigma }_t}\Gamma (r,t)d{\mu }_t={\Gamma }_0{\int }_{r_h\left(M\left(t\right)\right)}^{r_{\max }}\mathcal{G}(r;M\left(t\right))\sqrt{g_{rr}}dr.$$

(A30)

With our metric,

$$\sqrt{g_{rr}}=f{(r;M)}^{-1/2}={\displaystyle \frac{\ell }{\sqrt{r^2-r_h^2}}}.$$

Substituting $\mathcal{G}(r;M)={(r_h/r)}^p$ and performing the integral gives (letting $x=r/r_h$)

$$\begin{array}{cc}\hfill {\Gamma }_{\mathrm{eff}}\left(t\right)& ={\Gamma }_0{r}_h{\left(M\left(t\right)\right)}^{1-p}\ell {\int }_1^{x_{max}\left(t\right)}{\displaystyle \frac{x^{-p}}{\sqrt{x^2-1}}}dx,\hfill \end{array}$$

(A31)

where $x_{max}=r_{\max }/r_h\left(M\left(t\right)\right)$.

Horizon-dominated regime.

If the bath cutoff is large, $x_{max}\gg 1$, the integral converges for $p<1$ and can be evaluated in closed form using Beta functions. The crucial dependence on the horizon radius is the prefactor $\propto r_h^{1-p}$. Thus, using $r_h\propto M$ (JT scaling), we find

$${\Gamma }_{\mathrm{eff}}\left(t\right)\propto {\Gamma }_{0}M{\left(t\right)}^{1-p}.$$

Inserting the evaporation law $dM/dt=-\mu M^{-\gamma }$ and integrating yields

$$M\left(t\right)={\left(M_0^{1+\gamma }-(1+\gamma )\mu t\right)}^{1/(1+\gamma )}.$$

Therefore

$${\Gamma }_{\mathrm{eff}}\left(t\right)\propto {\Gamma }_0{\left(M_0^{1+\gamma }-(1+\gamma )\mu t\right)}^{{\displaystyle \frac{1-p}{1+\gamma }}}.$$

(A32)

Finite-time divergence and critical exponents.

Equation (A21) shows that ${\Gamma }_{\mathrm{eff}}\left(t\right)$ diverges at the finite time

$$t_{*}={\displaystyle \frac{M_0^{1+\gamma }}{(1+\gamma )\mu }}$$

if and only if the exponent $(1-p)/(1+\gamma )$ is negative, i.e. if $p>1$. For $p\le 1$ the integrated rate remains finite as $t\to t_{*}$. Thus in this toy model horizon-localization strong enough that $p>1$ is required to drive a blow-up of ${\Gamma }_{\mathrm{eff}}$ in finite evaporation time.

Appendix C.4. Interpretation and Relation to Page Time

The model is deliberately simple, but it exhibits three robust features:

1.

If the radial profile $\mathcal{G}$ concentrates sufficiently strongly near the horizon (large p), the integrated rate can diverge in finite time for a wide class of evaporation laws.

2.

If ${\Gamma }_{\mathrm{eff}}\left(t\right)$ grows but remains integrable up to the Page time $t_{\mathrm{Page}}$, semiclassical decoherence increases but does not necessarily lead to complete mixing before the Page time.

3.

The precise relation between ${\Gamma }_{\mathrm{eff}}$ and Page-time physics depends on microscopic constants (${\Gamma }_0,\mu ,p,\gamma$) and on the chosen profile $\mathcal{G}$. For four-dimensional Schwarzschild evaporation one expects different scaling exponents (e.g. $M^{-2}$ in certain regimes) and so a different condition for finite-time blow-up.

Appendix C.5. Caveats and Improvements

This appendix presents a toy JT-type calculation that illustrates how a time-dependent horizon radius induces time dependence of ${\Gamma }_{\mathrm{eff}}$. A more detailed calculation would:

• derive $\mathcal{G}(r;M)$ from first principles (contractions of the graviton noise kernel with the matter stress tensor in the JT background);
• include backreaction of the matter flux on the geometry beyond a simple evaporation law;
• adapt the spatial cutoff and slicing to match the physically relevant bath coupling used in JT evaporation literature (see [18,19]).

Despite these simplifications, the model captures the key parametric dependence of ${\Gamma }_{\mathrm{eff}}$ on $M\left(t\right)$ and thereby supplies a useful worked example to accompany the purity-decay theorem.

Appendix D.1. Algebraic Setting and GNS Representation

Let $\mathcal{A}$ be the algebra of local observables generated by smeared field operators $\varphi \left(f\right)$, $f\in C_0^{\infty }\left(M\right)$, subject to the field equations and canonical commutation relations in the usual way. Choose a Hadamard state $\omega$ on $\mathcal{A}$ and construct the GNS representation $({\mathcal{H}}_{\omega },{\pi }_{\omega },{\Omega }_{\omega })$. The dense domain $\mathcal{D}\subset {\mathcal{H}}_{\omega }$ is taken to be the finite-particle space (finite linear combinations of smeared-field excitations over ${\Omega }_{\omega }$).

All local fields are operator-valued distributions $\varphi \left(x\right)$ acting on $\mathcal{D}$ in the sense that for $f\in C_0^{\infty }$ the smeared operator $\varphi \left(f\right)=\int f\left(x\right)\varphi \left(x\right)d\mu$ is well defined and maps $\mathcal{D}$ into itself.

Appendix D.2. Definition and Renormalization of Composite Operators

Composite operators such as $\varphi {\left(x\right)}^2$, $T_{\mu \nu }\left(x\right)$, or more generally $L{\left(x\right)}^2$, are defined by point-splitting relative to the Hadamard parametrix $H(x,x^{\prime })$. Concretely, for a scalar field we define

$$:{\varphi }^2\left(x\right):=\underset{x^{\prime }\to x}{lim}\left(\varphi \left(x\right)\varphi \left(x^{\prime }\right)-H(x,x^{\prime })\right),$$

where the limit is taken in the sense of distributions on test functions. More generally, for a local operator $L\left(x\right)$ constructed polynomially from $\varphi$ and its derivatives, $L{\left(x\right)}^2$ is defined by the analogous point-split prescription and subtraction of the appropriate Hadamard singularity.

The Hadamard condition guarantees that these subtractions remove universal UV singularities and produce well-defined distributions (see [31,43]). Ambiguities in the subtraction are local curvature polynomials multiplied by finite renormalization constants; we assume these are fixed by physical renormalization conditions.

Appendix D.3. Smeared Quadratic Forms and Operator Domains

Rather than attempting to interpret $L{\left(x\right)}^2$ as a pointwise operator, we work with smeared quadratic forms

$${\mathcal{Q}}_f\left(\psi \right):={\int }_{{\Sigma }_t}f\left(x\right)\langle \psi ,{\pi }_{\omega }\left(L{\left(x\right)}^2\right)\psi \rangle d{\mu }_t,f\in C_0^{\infty }\left({\Sigma }_t\right),$$

defined for $\psi \in \mathcal{D}$. These quadratic forms are finite due to the point-splitting renormalization and Hadamard regularity. Under standard microlocal estimates (wavefront-set bounds), the forms are closable and give rise to self-adjoint operators via the Friedrichs construction when the smeared kernel f is nonnegative.

Appendix D.4. GKLS Generator on Trace-Class Operators

We wish to interpret the curvature-dependent GKLS generator as an operator on the Banach space ${\mathcal{T}}_1\left({\mathcal{H}}_{\omega }\right)$ of trace-class operators. The generator is first defined on the dense domain of finite-rank trace-class operators (finite linear combinations of rank-one projectors built from $\mathcal{D}$ vectors). Using the smeared quadratic forms above, the local dissipator

$$\mathcal{D}\left[\rho \right]:={\int }_{{\Sigma }_t}\Gamma \left(x\right)\left(L\left(x\right)\rho L\left(x\right)-\frac{1}{2}\{L{\left(x\right)}^2,\rho \}\right)d{\mu }_t$$

is well defined on this domain. Under the assumption that ${\Gamma }_{\mathrm{eff}}$ is finite and that $L\left(f\right)$ is essentially self-adjoint on $\mathcal{D}$ for all $f\in C_0^{\infty }$, the dissipator extends to a (densely defined) bounded generator of a contractive completely positive semigroup on ${\mathcal{T}}_1$. A detailed functional-analytic proof uses the following facts:

(D1)

Smeared boundedness: for fixed compactly supported f the operator $L\left(f\right)$ is essentially self-adjoint and the map $f\mapsto L\left(f\right)$ is continuous in the nuclear topology on test functions.

(D2)

Integrability:${\int }_{{\Sigma }_t}\Gamma \left(x\right){\parallel L\left(x\right)\psi \parallel }^{2}d{\mu }_t$ is finite for all $\psi \in \mathcal{D}$ by (A2) and the smearing used in defining $\Gamma$ (see Appendix B).

(D3)

Generator closability: the quadratic form associated with the dissipator is closed on $\mathcal{D}$ and hence the generator is closable; its closure generates a strongly continuous contraction semigroup on ${\mathcal{T}}_1$ (Hille–Yosida theorem and Lindblad theory).

These properties ensure the GKLS equation defines a well-posed evolution on the space of trace-class operators under our assumptions.

Appendix D.5. Summary of Assumptions for the Purity Theorem

To restate compactly the assumptions used in the main theorems:

• Hadamard state and GNS representation (control of UV singularities);
• local operators $L\left(x\right)$ defined as operator-valued distributions on a common dense domain $\mathcal{D}$;
• point-splitting renormalization of $L{\left(x\right)}^2$ with fixed finite renormalization constants;
• smearing and integrability conditions guaranteeing finite ${\Gamma }_{\mathrm{eff}}$;
• Gaussian metric fluctuations and Markov/local approximation (Appendix A);
• the dissipator gives a closable generator on trace-class operators.

Under these assumptions the statements of the purity theorem and the GKLS generator are mathematically well posed. When the assumptions fail (e.g. in ultra-singular spacetimes lacking Hadamard regularity), the construction requires further renormalization or a different framework (algebraic QFT with suitable extensions), which is beyond the scope of this work.

References for this appendix: Hollands & Wald [31], Fewster & Verch [32], Parker & Toms [43], and reviews of algebraic/renormalization methods in curved spacetime.