Von Neumann Entropy

Von Neumann entropy is defined for a quantum state described by a density matrix rho as S(rho) = -Tr(rho log rho), where the logarithm is typically taken in base 2 (giving units of qubits) or base e (nats). For a pure state, the density matrix satisfies rho^2 = rho and all eigenvalues are zero except one, so S(rho) = 0. For a maximally mixed state on a d-dimensional Hilbert space, rho = I/d and S(rho) = log d. This makes von Neumann entropy a precise measure of the ignorance or mixedness encoded in a quantum state.

The connection to Shannon entropy is direct: if rho is diagonal in some basis with eigenvalues p_i, then S(rho) = -sum p_i log p_i, exactly the Shannon entropy of the probability distribution {p_i}. This means Von Neumann entropy reduces to classical Shannon entropy whenever the quantum state has no coherences (off-diagonal terms) in the chosen basis. The Von Neumann entropy is basis-independent, however, because it depends only on the eigenvalues of rho, so it captures something genuinely intrinsic to the quantum state rather than to any particular measurement.

For a bipartite quantum system in a pure state |psi_AB>, the von Neumann entropy of the reduced density matrix rho_A = Tr_B(|psi_AB><psi_AB|) is the standard measure of entanglement entropy between subsystems A and B. Because the global state is pure, S(rho_A) = S(rho_B), and the shared value quantifies how entangled the two parts are: zero for product states and log(min(d_A, d_B)) for maximally entangled states. This makes entanglement entropy the central quantity in quantum information theory and condensed matter physics alike.

In many-body physics, entanglement entropy satisfies remarkable scaling laws known as area laws: for gapped local Hamiltonians in d spatial dimensions, the entanglement entropy of a region scales as the surface area of that region rather than its volume. Critical systems described by conformal field theory violate this with a logarithmic correction proportional to the central charge. Numerically, von Neumann entropy is computed by diagonalizing rho to obtain its eigenvalues {lambda_i} and evaluating S = -sum lambda_i log lambda_i, making eigenvalue decomposition the practical workhorse for entanglement calculations in quantum simulation and quantum chemistry.