What is "von neumann entropy"

n. (context quantum physics English) The entropy of a quantum state. If the state is expressed as a quantum density matrix $rho$, then this entropy can be expressed mathematically as $S\; =\; -text\{tr\}(rho\; log\; rho)$ where $text\{tr\}$ is the trace operator and the logarithm is natural logarithm.

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

S =  − tr(ρlnρ), 
where tr denotes the trace and ln denotes the (natural) matrix logarithm. If is written in terms of its eigenvectors |1〉, |2〉, |3〉, ... as

$$\rho = \sum_j \eta_j \left| j \right\rang \left\lang j \right| ~,$$
then the von Neumann entropy is merely

S =  − ∑ηlnη.
In this form, S can be seen to amount to the information theoretic Shannon entropy.