Entanglement entropy

Entanglement entropy is a measure of the quantum correlations between two subsystems of a composite quantum state. For a pure state, the entanglement entropy of a subsystem is the von Neumann entropy of its reduced density matrix, quantifying how much information is lost when the subsystem is considered in isolation. In the context of the holographic principle, entanglement entropy in a boundary quantum field theory corresponds to the area of extremal surfaces in the bulk gravitational geometry — a connection made precise by the Ryu-Takayanagi formula.

The systems-theoretic significance of entanglement entropy is that it provides a bridge between quantum information and geometry. In AdS/CFT correspondence, the entanglement structure of the boundary theory literally builds the bulk spacetime. Regions of high entanglement entropy correspond to deep bulk regions; disentangling boundary degrees of freedom causes bulk geometry to disconnect. Entanglement is not merely a quantum curiosity — it is the fabric from which space itself may be woven.