Von Neumann entropy

In physics, the von Neumann entropy, named after John von Neumann, is an extension of the concept of Gibbs entropy from classical statistical mechanics to quantum statistical mechanics. For a quantum-mechanical system described by a density matrix ρ, the von Neumann entropy is

${\displaystyle S=-\operatorname {tr} (\rho \ln \rho ),}$

where ${\displaystyle \operatorname {tr} }$ denotes the trace and ln denotes the (natural) matrix 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 information theoretic Shannon entropy.

The von Neumann entropy is also used in different forms (conditional entropies, relative entropies, etc.) in the framework of quantum information theory to characterize the entropy of entanglement.