Von Neumann entropy

In physics, the von Neumann entropy, named after John von Neumann, is a measure of the statistical uncertainty within a description of a quantum system. It extends the concept of Gibbs entropy from classical statistical mechanics to quantum statistical mechanics, and it is the quantum counterpart of the Shannon entropy from classical information theory. For a quantum-mechanical system described by a density matrix ρ, the von Neumann entropy is^[1] $${\displaystyle S=-\operatorname {tr} (\rho \ln \rho ),}$$ where ${\displaystyle \operatorname {tr} }$ denotes the trace and ${\displaystyle \operatorname {ln} }$ denotes the matrix version of the natural logarithm. If the density matrix ρ is written in a basis of its eigenvectors ${\displaystyle |1\rangle ,|2\rangle ,|3\rangle ,\dots }$ as $${\displaystyle \rho =\sum _{j}\eta _{j}\left|j\right\rangle \left\langle j\right|,}$$ then the von Neumann entropy is merely $${\displaystyle S=-\sum _{j}\eta _{j}\ln \eta _{j}.}$$ In this form, S can be seen as the Shannon entropy of the eigenvalues, reinterpreted as probabilities.^[2]

The von Neumann entropy and quantities based upon it are widely used in the study of quantum entanglement.^[3]