Visualize MPS tensor network structure, entanglement entropy S=−Σλᵢ²log(λᵢ²) from Schmidt decomposition, and how bond dimension χ controls entanglement. Compare area law vs volume law states.
MPS: |ψ⟩ = Σ Tr(A^{s₁}A^{s₂}...A^{sN})|s₁s₂...sN⟩. Bond dimension χ limits entanglement: S≤log(χ) (area law). Ground states of gapped 1D Hamiltonians obey area law (Hastings 2007) → efficiently representable as MPS with finite χ. DMRG (White 1992) implicitly optimizes an MPS. Critical systems have S∝log(L) (volume scaling) — CFT central charge c: S=(c/3)log(L).