Von Neumann Entropy

Von Neumann entropy is defined for a quantum state described by a density matrix $\rho$ as $S(\rho) = -\mathrm{Tr}(\rho \log \rho)$, 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 $S(\rho) = -\sum_i p_i \log p_i$, 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 $|\psi_{AB}\rangle$, the von Neumann entropy of the reduced density matrix $\rho_A = \mathrm{Tr}_B(|\psi_{AB}\rangle\langle\psi_{AB}|)$ 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 $S = -\sum_i \lambda_i \log \lambda_i$, making eigenvalue decomposition the practical workhorse for entanglement calculations in quantum simulation and quantum chemistry.