Entanglement and Geometric Relations

Entanglement can constrain geometry when a state, region, operator algebra, and entropy–area relation are specified. The central calculational bridge is the entanglement first law, a linear identity for nearby density operators. Turning it into a gravitational field equation requires extra geometric and dynamical assumptions; it is not a theorem that arbitrary entanglement produces spacetime.

Required background. Quantum Information supplies reduced density operators, relative entropy, modular Hamiltonians, and entropy inequalities.

Helpful background. Flat-Spacetime Quantum Field Theory supplies local stress tensors and vacuum regions; Einstein Equations supplies the target equation; Quantum Gravity supplies evidential boundaries; and Causality and Global Structure supplies domains of dependence and causal diamonds.

The entanglement first law

Begin with a finite-dimensional system, or a regulated region represented by density operators. Let $\rho_0$ be a faithful reference density operator (all its eigenvalues are positive), and define its modular Hamiltonian by

$$K_0=-\ln\rho_0.$$

For $\rho=\rho_0+\delta\rho$ with $\operatorname{Tr}\delta\rho=0$, vary $S(\rho)=-\operatorname{Tr}(\rho\ln\rho)$. To first order,

$$\delta S =-\operatorname{Tr}\!\left[\delta\rho(\ln\rho_0+1)\right] =\operatorname{Tr}(\delta\rho K_0) =\delta\langle K_0\rangle.$$

This equality is kinematic. It follows from normalization and differentiability of entropy, independently of gravity. Relative entropy supplies an independent check:

$$S(\rho\Vert\rho_0) =\Delta\langle K_0\rangle-\Delta S\ge0.$$

Because relative entropy begins at quadratic order around $\rho_0$, its linear term vanishes, reproducing the first law. Its quadratic term measures distinguishability beyond the linear identity. Identifying that term with a gravitational canonical energy requires an additional holographic or geometric construction; it is not true for an arbitrary density operator.

For a continuum QFT region the local algebra is generally of type III, so it does not have an ordinary reduced density matrix or finite von Neumann entropy. The displayed trace calculation is then a regulated derivation: meaningful continuum statements use algebraic relative entropy or finite variations after a common subtraction. The regulator and region algebra must agree on both sides of the first law.

From modular energy to curvature

For a ball of radius $R$ in the vacuum of a conformal field theory on flat spacetime, the modular Hamiltonian is local:

$$K_0=2\pi\int_{r\lt R}d^{d-1}x\, \frac{R^2-r^2}{2R}\,T_{00}(x).$$

Thus $\delta\langle K_0\rangle$ becomes a weighted stress-energy variation. A geometric argument must supply the other side: an entropy whose leading variation is an area variation, plus any bulk or matter entropy. In entanglement-equilibrium formulations, one holds the small causal diamond’s volume fixed and assumes the vacuum entropy is stationary. Expanding the area deficit in Riemann normal coordinates makes it proportional to the local Einstein tensor. Equating the geometric variation with the matter modular energy yields the linearized Einstein equation, with the cosmological constant entering as an integration ambiguity Jacobson 2016, pp. 1–4.

In a holographic setting, extremal-area entropy and the boundary first law similarly imply linearized bulk field equations for restricted semiclassical states Faulkner et al. 2014, §§ 2–4. Here that result is used only as an example of the logical chain; explicit duality constructions belong to the string-theory treatment.

Assumptions that carry the result

The derivation requires more than entanglement:

• a reference vacuum or equilibrium state and a controlled class of small perturbations;
• a region whose modular Hamiltonian is known or approximated;
• a factorization or algebra that defines the entropy, including treatment of gauge constraints and edge terms;
• a universal relation between geometric entropy and area;
• a local continuum regime in which curvature varies slowly across the region;
• the same metric in the area, stress-energy conservation, and causal-domain construction.

Changing any item can change the conclusion. Generic regions have nonlocal modular Hamiltonians. Gauge theories may not factorize into inside and outside Hilbert spaces without specifying an algebra or edge-mode extension. Higher-curvature gravity replaces area by a generalized entropy functional. Far-from-equilibrium states invalidate the linear variation.

The derivation also establishes an equation in a restricted state neighborhood, not a microscopic origin of the Hilbert space. It can show that a proposed entropy law is dynamically compatible with linearized gravity without explaining why that law or state is selected.

Distance from correlations

Mutual information,

$$I(A:B)=S(A)+S(B)-S(AB),$$

is nonnegative and bounds connected correlations. One may use its decay to infer adjacency: strongly correlated factors are treated as near. But a monotone relation between $I(A:B)$ and distance is state dependent. Critical states, topological order, and long-range interactions can sustain correlations without ordinary geometric proximity; product states can erase correlation on an existing geometry.

A reconstruction becomes stronger if correlation-derived distance agrees with propagation time, spectral response, and minimal-cut area over a scale range. Agreement must persist under small state changes and changes of microscopic representation. Correlations are then evidence for geometry, not simply a definition chosen to make the graph geometric.

Limits and independent checks

The most direct check is conservation. The inferred linearized Einstein equation must be compatible with $\nabla^\mu\delta T_{\mu\nu}=0$ through the linearized Bianchi identity. A sign check follows from positive energy in a small ball: positive modular energy increases matter entropy, while the fixed-volume geometric area variation must carry the compensating sign at equilibrium. A nonlinear claim needs relative entropy beyond first order and cannot be obtained by iterating the linear identity without controlling backreaction.

Exercise

Let $\rho_0=\operatorname{diag}(p,1-p)$ and $\delta\rho=\operatorname{diag}(\varepsilon,-\varepsilon)$. Verify the entanglement first law to first order and identify the quadratic relative-entropy term.

Solution

Expanding the binary entropy gives

$$\delta S =\varepsilon\ln\frac{1-p}{p}+O(\varepsilon^2).$$

Since $K_0=\operatorname{diag}(-\ln p,-\ln(1-p))$,

$$\delta\langle K_0\rangle =\varepsilon[-\ln p+\ln(1-p)] =\varepsilon\ln\frac{1-p}{p}.$$

The linear terms agree. Direct expansion gives

$$S(\rho_0+\delta\rho\Vert\rho_0) =\frac{\varepsilon^2}{2p(1-p)}+O(\varepsilon^3)\ge0.$$

The positive quadratic term is additional information; the first law alone only says that the relative entropy has no linear term.

References

• Faulkner, Thomas, Monica Guica, Thomas Hartman, Robert C. Myers, and Mark Van Raamsdonk (2014), “Gravitation from Entanglement in Holographic CFTs,” Journal of High Energy Physics 2014(3), 051, arXiv:1312.7856.
• Jacobson, Ted (2016), “Entanglement Equilibrium and the Einstein Equation,” Physical Review Letters 116, 201101, arXiv:1505.04753.