Von Neumann entropy

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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^[1]

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

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^[1]

${\displaystyle S=-\sum _{j}\eta _{j}\ln \eta _{j}.}$

In this form, S can be seen as the information theoretic Shannon entropy.^[1]

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.^[2]