vonNeumannEntropie

Von Neumann entropy is a measure of quantum uncertainty in a state described by a density operator ρ on a Hilbert space. It is defined as S(ρ) = - Tr(ρ log ρ). Equivalently, if ρ has eigenvalues λ_i, then S(ρ) = - Σ_i λ_i log λ_i. The entropy vanishes for a pure state and reaches its maximum log d for the maximally mixed state in a d-dimensional Hilbert space. The logarithm base determines units (natural logarithm gives nats; base 2 gives bits).

Key properties: unitary invariance S(UρU†) = S(ρ); S is a concave function of ρ; it is nonincreasing under completely positive trace-preserving maps (data processing inequality); subadditivity S(ρAB) ≤ S(ρA) + S(ρB); strong subadditivity S(ρABC) + S(ρB) ≤ S(ρAB) + S(ρBC).

In bipartite systems, the von Neumann entropy of each marginal equals the entanglement entropy for pure states: S(ρA) = S(ρB) and measures the amount of entanglement between subsystems. It also plays a role in quantum thermodynamics: for thermal states ρ ∝ e^{-βH}, S relates to thermodynamic entropy, heat and temperature. In practice, entropy can be estimated from tomography or bounded through interferometric or correlation measurements.

Historically, the concept generalizes Shannon entropy; it was introduced by John von Neumann. In German literature the term is usually written as Von Neumann-Entropie. The terminology is standard in quantum information theory, and the quantity is central to questions of information, entanglement, and state distinguishability.