Editing Entropy (section)

=== Entropy in quantum mechanics ===
{{Main|von Neumann entropy}}

In [[quantum statistical mechanics]], the concept of entropy was developed by [[John von Neumann]] and is generally referred to as "[[von Neumann entropy]]":<math display="block">S = - k_\mathsf{B}\ \mathrm{tr}{\left( \hat{\rho} \times \ln{\hat{\rho}} \right)}</math>where <math display="inline">\hat{\rho}</math> is the [[density matrix]], <math display="inline">\mathrm{tr}</math> is the [[Trace class|trace operator]] and <math display="inline">k_\mathsf{B}</math> is the [[Boltzmann constant]].

This upholds the [[correspondence principle]], because in the [[classical limit]], when the phases between the basis states are purely random, this expression is equivalent to the familiar classical definition of entropy for states with classical probabilities <math display="inline">p_i</math>:<math display="block">S = - k_\mathsf{B} \sum_i{p_i \ln{p_i}}</math>i.e. in such a basis the density matrix is diagonal.

Von Neumann established a rigorous mathematical framework for quantum mechanics with his work {{lang|de|Mathematische Grundlagen der Quantenmechanik}}. He provided in this work a theory of measurement, where the usual notion of [[wave function collapse]] is described as an irreversible process (the so-called von Neumann or [[projective measurement]]). Using this concept, in conjunction with the [[density matrix]] he extended the classical concept of entropy into the quantum domain.