Geometry as compression in theoretical physics

What work does a picture save?

Given a quantum state and a spatial region, computing entanglement entropy involves a reduced density matrix. Under specific conditions, holography turns this task into a geometric extremization problem: fix the boundary region, find surfaces satisfying the boundary and homology conditions, and compute their areas. Once a curve is drawn, its state, region and entropy prescription still need to be specified.

“Compression” here concerns computational relations. A few distances, connection patterns and extremality conditions collect information previously spread across equations. Testing the representation requires retaining a dictionary from the picture back to the original observable.

A single-interval example

Consider a two-dimensional conformal field theory in its zero-temperature vacuum on an infinite line. The region is a single interval of length L; ε is an ultraviolet cutoff much smaller than L, and c is the central charge. With natural logarithms, its entropy is:

The constant s₀ depends on the regulator convention. Comparing the two lengths in the same theory, state and cutoff cancels it. For a theory with a classical Einstein-AdS₃ dual, the RT prescription expresses the leading entropy as a geodesic length divided by 4G₃, where G₃ is the three-dimensional Newton constant. On a fixed-time Poincaré slice, the geodesic for a single interval is a semicircle. Integrating along it and matching cutoffs gives the logarithmic dependence above.

The semicircle brings together endpoint anchoring, minimal length and ultraviolet divergence. Its outline alone does not specify c or s₀; the gravitational coupling and regulator dictionary supply that information.

How multiple intervals enter an optimization

Multiple intervals require comparing connection patterns. The public preprint Information clustering and geometric pressure in AdS3 goes on to study grouped storage: correlations save memory, while merging groups incurs a nonnegative cost.

Its storage objective is defined at a fixed finite factorizing regulator. For fixed finitely many separated intervals, the leading Einstein-AdS₃ vacuum RT entropy model gives a representation through free-perimeter-plus-area optimization, with a controlled ultraviolet error. Using actual quantum-state entropies requires separate error bounds. The compact-BTZ annulus has a finite mismatch at the unchanged tariff. These conditions determine how far the geometric representation applies.

Turn the picture back into checks

1. Write down the inputs: state, interval endpoints, boundary conditions, cutoff and entropy definition.
2. For a single interval, calculate the entropy difference on doubling its length using the field-theory formula. Check the geodesic result with the same cutoff dictionary.
3. For multiple intervals, list and evaluate candidate connections satisfying homology. Drawing only one appealing connection can omit a competing solution.
4. Recheck the dictionary when changing the model. Finite temperature, finite system size and quantum corrections can all change the calculation.

The single-interval CFT formula and the classical holographic geometry have distinct domains of applicability. The former does not require every CFT to possess a classical gravity dual; using the latter requires stating the classical-gravity approximation. Keeping these inputs beside the picture gives the reader a route to reproduce the calculation.

Original references

• P. Calabrese and J. Cardy, Entanglement Entropy and Quantum Field Theory, J. Stat. Mech. (2004) P06002, arXiv:hep-th/0405152. Single-interval vacuum and finite-temperature results.
• S. Ryu and T. Takayanagi, Holographic Derivation of Entanglement Entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602, arXiv:hep-th/0603001. See equations (2.10)–(2.11) in Poincaré coordinates.
• F. Deng, Information clustering and geometric pressure in AdS3, public Zenodo preprint v1 (2026), DOI:10.5281/zenodo.23534563. Geometric representation of information partitions and its conditions.

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