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 , the von Neumann entropy is where denotes the trace and ln denotes the (natural) matrix logarithm. If the density matrix is written in a basis of its eigenvectors as then the von Neumann entropy is merely 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. Background John von Neumann established a rigorous mathematical framework for quantum mechanics in his 1932 work Mathematical Foundations of Quantum Mechanics. In it, he provided 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). The density matrix was introduced, with different motivations, by von Neumann and by Lev Landau. The motivation that inspired Landau was the impossibility of describing a subsystem of a composite quantum system by a state vector. On the other hand, von Neumann introduced the density matrix in order to develop both quantum statistical mechanics and a theory of quantum measurements. The density matrix formalism, thus developed, extended the tools of classical statistical mechanics to the quantum domain. In the classical framework, the probability distribution and partition function of the system allows us to compute all possible thermodynamic quantities. Von Neumann introduced the density matrix to play the same role in the context of quantum states and operators in a complex Hilbert space. The knowledge of the statistical density matrix operator would allow us to compute all average quantum entities in a conceptually similar, but mathematically different, way. Let us suppose we have a set of wave functions |Ψ〉 that depend parametrically on a set of quantum numbers n1, n2, ..., nN. The natural variable which we have is the amplitude with which a particular wavefunction of the basic set participates in the actual wavefunction of the system. Let us denote the square of this amplitude by p(n1, n2, ..., nN). The goal is to turn this quantity p into the classical density function in phase space. We have to verify that p goes over into the