Matrix Product States & Entanglement Area Law

In 1D quantum systems, the entanglement entropy of a subsystem scales with the boundary size (area law), not the volume. Matrix product states (MPS) capture this by representing quantum states with a chain of tensors of bond dimension χ — larger χ means more entanglement can be represented.

Parameters

Bond dimension χ: 4 System size N: 12 Bipartition cut: 6 Critical exponent c: 1.0

S(ℓ) = (c/3) ln[sin(πℓ/N)] + const

Area law: S ~ const for gapped systems
Critical: S ~ (c/3) ln ℓ (CFT prediction)
Bond dim χ truncates Schmidt values: S ≤ ln(χ)