|ψ⟩ = ∑ A[1]ⁱ¹A[2]ⁱ²···A[N]ⁱᴺ |i₁i₂···iN⟩ | Area Law
S(A) = —S_max = —Ratio S/S_max = —DOF: —
Matrix Product States (MPS) are tensor network ansätze that efficiently represent
ground states of gapped 1D Hamiltonians. The key parameter is the bond dimension χ:
entanglement entropy across any cut is bounded by S ≤ log₂(χ) — this is the area law
(area = a single point in 1D). In contrast, generic quantum states (Page states) satisfy the
volume law S ≈ N/2 log₂(d). DMRG (White 1992) implicitly works with MPS; the modern
tensor network formulation (Vidal 2003) makes this explicit. For critical systems described
by CFT, the entanglement grows logarithmically: S = (c/3) log(N/π sin(πℓ/N)).
Visualization shows the MPS network, singular value spectra at the cut, and entanglement entropy profile.