Entanglement Entropy

To define entanglement entropy, first partition the total Hilbert space into two subsystems A and B, so the total state lives in H_A tensor H_B. The reduced density matrix of subsystem A is obtained by taking the partial trace over B: rho_A = Tr_B(rho). This operation discards the degrees of freedom of B and leaves a description of A alone. If the total state is a product state |psi_A> tensor |psi_B>, then rho_A is a pure state and Tr_B does nothing surprising. If the total state is entangled, however, rho_A is a mixed state even though the global state may be pure, and the mixedness of rho_A is a direct signature of entanglement between A and B.

The entanglement entropy is then S(A) = -Tr(rho_A log rho_A), the von Neumann entropy of the reduced density matrix. Equivalently, if the singular values of the Schmidt decomposition of |psi> are lambda_i, then S(A) = -sum_i lambda_i^2 log(lambda_i^2). For a product state all weight lies on one Schmidt vector, giving S = 0. For a maximally entangled state of local dimension d, the Schmidt values are all 1/sqrt(d) and S reaches its maximum value of log(d). Entanglement entropy is symmetric: S(A) = S(B) for any pure global state, even when the dimensions of A and B differ.

One of the deepest results connecting entanglement entropy to physics is the area law versus volume law distinction for ground states of local Hamiltonians. Gapped local Hamiltonians in one dimension satisfy an area law: the entanglement entropy across any bipartition scales with the boundary area (a constant in 1D), not the volume of the subsystem. Critical systems and systems with long-range interactions can exhibit logarithmic corrections or volume-law scaling. This distinction is directly connected to the efficiency of tensor network methods: states satisfying area laws can be efficiently represented as matrix product states or projected entangled pair states, while volume-law states require exponentially large tensors and cannot be efficiently simulated classically.

Entanglement entropy therefore serves as a measure of quantum simulation hardness: the higher the entanglement entropy across a bipartition, the harder the state is to represent and time-evolve classically. This is why quantum advantage in simulation is expected to appear first for strongly correlated materials and quantum chaos, where entanglement entropy grows rapidly under dynamics. The connection to quantum error correction is complementary: a good quantum error-correcting code is precisely a subspace of the Hilbert space where logical information is highly nonlocally encoded, meaning the entanglement entropy of any small subsystem is maximal. High entanglement, which is a bug for classical simulation, is a feature for error correction.