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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
where tr {\displaystyle \operatorname {tr} } denotes the trace and ln denotes the matrix logarithm. If ρ is written in terms of its eigenvectors | 1 ⟩ , | 2 ⟩ , | 3 ⟩ , … {\displaystyle |1\rangle ,|2\rangle ,|3\rangle ,\dots } as
then the von Neumann entropy is merely
In this form, S can be seen as the information theoretic Shannon entropy.
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