From Quantum Relative Entropy to the Semiclassical Einstein Equations

This paper demonstrates that the semiclassical Einstein equations naturally emerge from the proportionality between quantum relative entropy and horizon area variations, thereby generalizing Jacobson's thermodynamic derivation by replacing classical entropy with well-defined quantum information measures.

The Gist

Imagine you are standing in a vast, empty room. This room represents the vacuum of space—nothing but "empty" space. Now, imagine you throw a single pebble into this room. The air ripples, the light bends, and the room feels different.

For decades, physicists have known that matter (the pebble) and gravity (the bending of the room) are deeply connected. Einstein's famous equations describe exactly how matter tells space how to curve, and how space tells matter how to move. But why does this connection exist? Is it just a fundamental rule of the universe, or is there a deeper reason?

In this paper, two physicists from Leipzig, Germany, propose a fascinating answer: Gravity is actually a form of information processing.

Here is the story of their discovery, broken down into simple concepts.

1. The "Thermodynamics" of Space

In the 1990s, a physicist named Ted Jacobson had a brilliant idea. He suggested that Einstein's equations aren't just about geometry; they are about thermodynamics (the science of heat and energy).

He imagined a tiny, invisible wall in space (called a "Rindler horizon"). If you shine heat (energy) at this wall, the wall's "entropy" (a measure of disorder or information) changes. Jacobson showed that if you assume the area of this wall changes in a specific way when heat hits it, you can mathematically derive Einstein's equations.

The Problem: Jacobson's argument relied on "classical" thermodynamics. But space isn't just a smooth sheet; it's made of quantum particles. In the quantum world, the usual way we measure "disorder" (called von Neumann entropy) breaks down and gives infinite, nonsensical numbers.

2. The New Tool: "Relative Entropy"

The authors of this paper say, "Let's fix this by using a better tool."

Instead of measuring the total disorder of the room, they measure the difference between two states.

• State A: The empty room (the vacuum).
• State B: The room with the pebble thrown in (a "coherent excitation").

In the language of quantum information, this difference is called Relative Entropy. Think of it like a "distinguishability score."

• If the room is empty and you look at it, it looks exactly like the empty room. The score is zero.
• If you throw a pebble in, the room looks different. The score goes up.

The authors prove that in the quantum world, this "distinguishability score" is perfectly well-defined and finite, even where other methods fail.

3. The Magic Connection: Information = Energy

Here is the core magic of their paper. They used advanced math (called "Modular Theory") to look at what happens at the edge of this local "room" (the horizon).

They discovered a direct link:

The "distinguishability score" (Relative Entropy) is exactly equal to the amount of energy flowing across the horizon.

Imagine you are watching a river.

• Old View: You see water flowing (Energy).
• New View: The authors say the amount of water flowing is actually a measure of how much the "information" of the river has changed compared to a calm, still river.

If you add energy (throw a pebble), the "information difference" between the calm river and the rippling river increases. The math shows these two things are two sides of the same coin.

4. From Information to Gravity

Now, they bring in the famous Bekenstein-Hawking idea: The entropy of a black hole (or a horizon) is proportional to its surface area.

So, the chain of logic becomes:

1. Energy flows across the horizon.
2. This energy is mathematically identical to a change in Relative Entropy (Information).
3. If we assume that Information is proportional to Area (just like in black hole physics), then:
   • Energy Flow $\rightarrow$ Change in Area.

When you translate "Change in Area" back into the language of geometry, you get Einstein's Equations.

The Big Picture: Why This Matters

This paper suggests that gravity isn't a fundamental force like magnetism. Instead, it is an emergent phenomenon, much like temperature.

• Temperature isn't a property of a single atom; it's a statistical result of how trillions of atoms move.
• Gravity might not be a fundamental property of space; it might be the result of how quantum information changes when matter is present.

The Analogy:
Imagine a crowd of people in a stadium.

• If everyone is standing still, the "information" of the crowd is low.
• If a wave starts (people standing up and sitting down), the "information" changes.
• The authors are saying that the "wave" (gravity) is just the stadium's way of reacting to the change in information caused by the people moving.

Conclusion

The authors have taken a complex thermodynamic argument and upgraded it with modern quantum information theory. They showed that if you look at space through the lens of quantum information (specifically, how distinguishable an excited state is from the vacuum), the laws of gravity fall out naturally.

It suggests that at the deepest level of reality, space, time, and gravity are woven together by the fabric of information.

Technical Summary

Here is a detailed technical summary of the paper "From Quantum Relative Entropy to the Semiclassical Einstein Equations" by Philipp Dorau and Albert Much.

1. Problem Statement

The paper addresses the foundational question of whether the Einstein field equations can be derived entirely within the framework of Quantum Field Theory (QFT), without relying on classical thermodynamic analogies or semi-classical assumptions that are not rigorously defined in QFT.

• Context: Ted Jacobson (1995) famously derived the Einstein equations by applying the thermodynamic relation $\delta Q = T \delta S$ to local Rindler horizons. However, his derivation relies on the concept of entropy ($\delta S$) and heat flux ($\delta Q$) in a way that is difficult to define rigorously in QFT due to ultraviolet (UV) divergences.
• The Obstacle: In local QFT, the von Neumann entropy is typically ill-defined because local algebras are of Type III, leading to infinite entanglement across boundaries.
• The Goal: To formulate a derivation of the semiclassical Einstein equations using Quantum Relative Entropy, a well-defined quantity in QFT that quantifies the distinguishability between quantum states, thereby replacing classical thermodynamic entropy with a rigorous information-theoretic measure.

2. Methodology

The authors employ Algebraic Quantum Field Theory (AQFT) and Modular Theory (Tomita-Takesaki theory) to construct the derivation. The strategy proceeds in the following steps:

A. Local Spacetime Geometry

• Equivalence Principle: They approximate a small neighborhood $U$ of a spacetime manifold $M$ as flat Minkowski space.
• Local Rindler Horizon: Within this region, they consider a uniformly accelerated observer, generating a bifurcate Killing horizon (consisting of two null hypersurfaces $H_A$ and $H_B$ intersecting at a 2-surface $S$). This mimics the local Rindler horizon used in Jacobson's original work.

B. Algebraic Setup and States

• Field: A real, minimally coupled scalar field $\Phi$ satisfying the Klein-Gordon equation.
• States:
  • Vacuum State ($\omega_0$): A quasifree, Hadamard state invariant under the Killing flow (analogous to the Minkowski vacuum).
  • Coherent Excitation ($\omega_\phi$): A state generated by a coherent excitation of the vacuum, defined via Weyl operators $W(\phi)$.
• Modular Theory: They utilize the modular group $\Delta_R^{it}$ associated with the von Neumann algebra of observables localized in the right wedge of the horizon. For the vacuum state, this modular group acts geometrically as affine dilations along the horizon.

C. Calculation of Relative Entropy

• The authors calculate the Araki-Uhlmann relative entropy $S_{rel}(\omega_0 \| \omega_\phi)$ between the vacuum and the coherent excitation.
• Using the geometric action of the modular operator, they derive an explicit expression for the relative entropy in terms of the symplectic form of the field on the horizon.
• Key Result: The relative entropy is shown to be directly proportional to the energy flux of the field across the horizon:
  $$S_{rel}(\omega_0 \| \omega_\phi) = -2\pi \int_{H_B^R} U \langle :T_{ab}: \rangle_{\omega_\phi} \xi^a \xi^b \, dU d\text{vol}_S$$
  where $\xi^a$ is the Killing vector, $U$ is the affine parameter, and $:T_{ab}:$ is the normal-ordered energy-momentum tensor.

D. Linking to Geometry

• Area Variation: They assume the Bekenstein-Hawking relation holds, positing that the relative entropy (information) is proportional to the variation in the horizon's cross-sectional area ($\delta A$).
• Raychaudhuri Equation: They relate the area variation $\delta A$ to the expansion scalar $\theta$ of null geodesics. Using the Raychaudhuri equation and the fact that shear and vorticity vanish on a Killing horizon, they relate $\delta A$ to the Ricci tensor component $R_{ab}\xi^a\xi^b$.

3. Key Contributions

1. Rigorous QFT Derivation: The paper provides a derivation of the semiclassical Einstein equations that is strictly rooted in QFT, avoiding the ill-defined nature of von Neumann entropy in local algebras by using Relative Entropy.
2. Identification of Energy Flux: It establishes a precise mathematical identity between the quantum relative entropy of coherent states and the classical energy flux across a local horizon. This bridges the gap between information theory and energy-momentum transport in curved spacetime.
3. Generalization of Jacobson's Work: It extends Jacobson's thermodynamic argument by replacing the heuristic $\delta Q = T \delta S$ with a rigorous quantum information relation: Information Distinguishability $\propto$ Energy Flux $\propto$ Area Variation.
4. Derivation of the Einstein Equations: By equating the information-theoretic flux (relative entropy) with the geometric flux (Ricci curvature via Raychaudhuri), the authors recover the full semiclassical Einstein equations.

4. Results

The derivation leads to the following logical chain:

1. Relative Entropy $\leftrightarrow$ Energy Flux:
   $$S_{rel} \propto \int U \langle :T_{ab}: \rangle \xi^a \xi^b$$
2. Entropy-Area Relation: Assuming $S_{rel} = \frac{1}{4} \delta A$ (analogous to Bekenstein-Hawking), the energy flux is proportional to the area variation.
3. Geometric Relation: The area variation is governed by the Raychaudhuri equation, linking it to the Ricci tensor:
   $$\delta A \propto \int U R_{ab} \xi^a \xi^b$$
4. Final Equation: Equating the two expressions for $\delta A$ yields:
   $$R_{ab} - \frac{1}{2}R g_{ab} + \Lambda g_{ab} = 8\pi \langle :T_{ab}: \rangle$$
   (where the proportionality constant $\alpha$ is fixed to $8\pi$ by the choice of the entropy-area coefficient).

The result confirms that the semiclassical Einstein equations emerge naturally from the requirement that the quantum information distinguishability (relative entropy) between vacuum and excited states is proportional to the geometric area variation of local horizons.

5. Significance

• Quantum Gravity Insights: The work suggests that spacetime curvature (gravity) is an emergent phenomenon arising from quantum information. Specifically, the "zeroth-order" approximation of quantum gravity (semiclassical gravity) is dictated by the distinguishability of quantum states on local horizons.
• Resolution of UV Divergences: By utilizing relative entropy, the authors bypass the UV divergence problems associated with von Neumann entropy in Type III algebras, offering a mathematically robust path for thermodynamic derivations of gravity.
• Coherent States as Matter: The paper interprets coherent excitations as the simplest model of infalling matter. The relative entropy quantifies the "entanglement deficit" caused by matter, linking the presence of matter directly to the focusing of light rays (curvature).
• Future Directions: The authors note that while the derivation is rigorous for coherent states, extending it to general states (non-coherent) and higher-order corrections (beyond the local Minkowski approximation) remains a critical avenue for future research, potentially connecting to the Quantum Null Energy Condition (QNEC).

In summary, Dorau and Much demonstrate that the Einstein equations are not merely a classical limit but a direct consequence of the fundamental quantum information structure of fields in curved spacetime.