Area Laws

Why do tensor networks work so well for physical systems? The deep reason is the area law for entanglement entropy. It explains, in one sentence, which states have small bond dimension: those whose entanglement scales with the boundary of a region rather than its volume.

Volume law vs. area law

Take a quantum state on a large system and cut out a region $A$ of $\ell$ qubits. The entanglement between $A$ and the rest is quantified by the entanglement entropy $S(A)$ , the von Neumann entropy of the reduced density matrix $\rho_A$ :

S(A) = -\operatorname{Tr}\bigl(\rho_A \log_2 \rho_A\bigr).

Math you'll use here — reading the matrix logarithm. The expression $-\operatorname{Tr}(\rho_A \log_2 \rho_A)$ looks like it needs a logarithm of a matrix, but you never have to compute one. Because $\rho_A$ is Hermitian and positive with unit trace, it has eigenvalues $p_i$ that are real, non-negative, and sum to one — exactly a probability distribution. Diagonalising $\rho_A = \sum_i p_i \lvert i\rangle\langle i\rvert$ turns the trace into a plain sum over those eigenvalues:

S(A) = -\sum_i p_i \log_2 p_i ,

the Shannon entropy of the eigenvalue distribution $\{p_i\}$ (with the convention $0\log_2 0 = 0$ ). So "von Neumann entropy" just means: find the eigenvalues of the reduced state, then take their Shannon entropy. A pure $\rho_A$ has one eigenvalue $1$ and the rest $0$ , giving $S(A)=0$ ; a maximally mixed $\rho_A$ on $\ell$ qubits has $2^\ell$ equal eigenvalues $2^{-\ell}$ , giving the volume-law value $S(A)=\ell$ . Reading $S(A)$ as eigenvalue spread is all the area law below relies on.

A generic (random) state obeys a volume law: $S(A)$ grows with the number of qubits inside $A$ , $S(A) \sim \ell$ . Such states are maximally entangled and genuinely need exponential resources to store.

A state obeys an area law when $S(A)$ instead scales with the size of the boundary of $A$ — the surface separating it from the rest:

S(A) \sim |\partial A| .

In one dimension a contiguous block has a boundary consisting of (at most) two endpoints, so the boundary has constant size and $S(A) = O(1)$ : the entanglement saturates and does not grow with $\ell$ . In two dimensions a region of linear size $L$ has a boundary of length $\sim L$ , so $S(A) \sim L$ rather than $L^2$ .

Why the area law makes tensor networks efficient

Recall from earlier in the module that the bond dimension across a cut must be at least $2^{S}$ to represent entanglement entropy $S$ exactly — equivalently $S \le \log_2 \chi$ . So:

1D area law ( $S = O(1)$ ): bond dimension is bounded by a constant independent of system size. The state has an efficient MPS — this is the rigorous foundation for why MPS and DMRG work.
2D area law ( $S \sim L$ ): bond dimension must scale like $2^{L}$ across a boundary of length $L$ , which is why faithful 2D tensor networks (PEPS) are far harder, as we will see.

Which states obey area laws?

Area laws are the rule for ground states of gapped, local Hamiltonians — the energetically lowest state of a system with short-range interactions and a finite energy gap above the ground state. These describe a huge fraction of condensed- matter systems at low temperature.

Area laws can be violated, and the violations are exactly the hard cases:

Critical (gapless) 1D systems show a logarithmic correction, $S \sim \tfrac{c}{6}\log \ell$ , where $c$ is the central charge. The bond dimension then grows polynomially with system size — still tractable, but no longer constant.
Highly excited or thermalising states typically obey a volume law and are not efficiently representable.
Deep random circuits generate volume-law entanglement, which is precisely why they resist classical simulation.

The takeaway

The area law sorts quantum states into "tensor-network easy" and "tensor-network hard" by how their entanglement scales with region size. Whenever you ask whether a state can be simulated cheaply, the first question to pose is: does its entanglement obey an area law, and in how many dimensions? The answer fixes how the required bond dimension grows, and therefore whether the simulation is feasible.