Quantum Entanglement: Area Law vs Volume Law

System Size L: Entanglement Range ξ: Ground State (Area Law) Excited State (Volume Law) Random State (Max Entanglement) MBL Phase (Area Law) Page Curve (Random) Animate

In 1D quantum systems, ground states of gapped Hamiltonians obey the area law: S(L_A) ≈ const. Critical ground states follow S ~ (c/3)log(L_A) (CFT). Excited/random states satisfy the volume law: S(L_A) ~ L_A. This sharp distinction underpins tensor network methods like MPS/DMRG.