Entanglement Entropy
Entanglement entropy quantifies how entangled a bipartite pure state is, as the von Neumann entropy of a subsystem's reduced state.
How much entanglement
Entanglement entropy measures the degree of entanglement between two parts of a pure quantum state. It is defined as the von Neumann entropy of the reduced density matrix of one part: S(rho_A) = -Tr(rho_A log2 rho_A). For a pure whole, S(rho_A) = S(rho_B).
Reading the number
- S = 0: product state, no entanglement (reduced state is pure)
- S between 0 and 1 bit: partial entanglement of two qubits
- S = 1 bit: maximal entanglement, e.g. a Bell state
The entropy equals the mixedness of the subsystem. A pure reduced state means no entanglement; a maximally mixed reduced state means maximal entanglement. So entanglement of a pure bipartite state is exactly the local uncertainty it induces.
Computing it
Diagonalise rho_A to get eigenvalues lambda_i, which are the squared Schmidt coefficients of the state. Then S = -sum_i lambda_i log2 lambda_i. For the Bell state the eigenvalues are 1/2 and 1/2, giving S = 1 bit — one full ebit of entanglement.
Schmidt decomposition
Any bipartite pure state can be written as sum_i sqrt(lambda_i) |a_i>|b_i> with orthonormal local bases. The number of nonzero terms (the Schmidt rank) tells whether the state is entangled at all, and the spread of the lambda_i sets the entropy. This decomposition makes entanglement entropy easy to compute and interpret.
Where it is used
Entanglement entropy is central well beyond computing: it characterises quantum phases of matter, bounds how efficiently tensor-network methods can classically simulate a system, and appears in black-hole thermodynamics. In quantum computing, low entanglement entropy often signals that a state is classically simulable, so high entanglement is a prerequisite for quantum advantage.