Entanglement Entropy

For a pure bipartite state, the von Neumann entropy of either subsystem is the definitive measure of how entangled the two parts are.

Definition

Split a pure state into parts A and B. The entanglement entropy is S(rho_A) = -Tr(rho_A log rho_A), where rho_A is the reduced density matrix of A. By the Schmidt decomposition S(rho_A) = S(rho_B), so the choice of which part to keep does not matter. In terms of Schmidt coefficients, S = -sum_i lambda_i^2 log lambda_i^2.

Range and meaning

The entropy vanishes for product states and is maximal, log d, for maximally entangled states of local dimension d. It has an operational meaning: it is the rate at which maximally entangled Bell pairs can be distilled from, or are needed to prepare, many copies of the state by local operations and classical communication. This makes it the unique measure of pure-state bipartite entanglement in the asymptotic setting.

import numpy as np
def ent_entropy(psi, dA, dB):
    M = psi.reshape(dA, dB)
    s = np.linalg.svd(M, compute_uv=False)
    p = s**2; p = p[p>1e-12]
    return -np.sum(p*np.log2(p))
bell = np.array([1,0,0,1])/np.sqrt(2)
print(ent_entropy(bell,2,2))   # 1 bit

Caution for mixed states

Entanglement entropy is a valid entanglement measure only for pure global states. For a mixed rho_AB the von Neumann entropy of the reduced state mixes classical and quantum correlations and does not isolate entanglement; one must turn to measures like negativity or entanglement of formation instead. Confusing the two is a common error.

Area laws and simulation

In many-body physics the way entanglement entropy scales with subsystem size governs simulability. Ground states of gapped local Hamiltonians typically obey an area law, with entropy scaling as the boundary rather than the volume; such states are efficiently captured by tensor networks. States with volume-law entropy, generic after chaotic evolution, resist classical description. Entanglement entropy is therefore both a physical diagnostic and a practical measure of computational hardness.