Holographic Entanglement Entropy

Boundary Region A

Interval size 4 sites

Interval start site 0

Bulk Tensor Network

Network depth 5 layers

Bond dimension log₂χ 2

Readouts

Boundary sites N

|A| / N

12.5%

Entropy S(A)

0.0 bits

Edges cut (RT surface)

How it works

A tensor network (here a MERA-style binary tree, one concrete model of the AdS/CFT correspondence) discretises the idea of "emergent spacetime": the boundary layer is a chain of N entangled qubits (a toy quantum system), and each layer above it is a fictitious "bulk" that geometrises the entanglement pattern of the layer below.

The Ryu–Takayanagi conjecture states that the entanglement entropy of a boundary region A equals the area of the minimal bulk surface homologous to A, in Planck units:

S(A) = Area(γ_A) / 4G_N

In a discrete tensor network this becomes a graph theorem (Bao, Pollack, Remmen 2015 — random tensor networks): the area is replaced by a minimal edge cut through the bulk graph separating region A's legs from the rest, each severed bond contributing log₂χ bits (χ = bond dimension):

S(A) = min-cut(A, Ā) · log₂χ   [max-flow / min-cut theorem]

This simulator builds the tree, then runs a real Edmonds–Karp max-flow from a super-source wired to A's boundary legs to a super-sink wired to the rest — the resulting min cut (red edges) is the discrete Ryu–Takayanagi surface.

Interval size / start — choose which contiguous block of boundary qubits is "region A".
Network depth — more layers ⇒ more boundary sites (N = 2^depth) and a deeper bulk.
Bond dimension χ — how much entanglement a single bulk bond can carry; scales S(A) linearly.

The signature holographic result you can see directly: S(A) grows with the boundary of the region (a near-constant number of cut edges once A is a small contiguous block), not with its volume — the same area-law scaling that makes black-hole entropy proportional to horizon area rather than interior volume.

Region A (boundary) Complement Ā RT surface (min cut)

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