Editing Open quantum system (section)

==Quantum system and environment==

A complete description of a quantum system requires the inclusion of the environment. Completely describing the resulting combined system then requires the inclusion of its environment, which results in a new system that can only be completely described if its environment is included and so on. The eventual outcome of this process of embedding is the state of the whole universe described by a [[wavefunction]] <math>\Psi</math>. The fact that every quantum system has some degree of openness also means that no quantum system can ever be in a [[pure state]].
[[File:System-bath.jpg|thumb|System bath partition]]
Even if the combined system is in a pure state and can be described by a wavefunction <math> \Psi </math>, a subsystem in general cannot be described by a wavefunction. This observation motivated the formalism of [[density matrices]], or density operators, introduced by [[John von Neumann]]<ref>{{Citation|last=von Neumann|first=John|title=Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik|journal=Göttinger Nachrichten|volume=1|pages=245–272|year=1927|authorlink=John von Neumann}}</ref> in 1927 and independently, but less systematically by [[Lev Landau]] in 1927 and [[Felix Bloch]] in 1946. In general, the state of a subsystem is described by the density operator <math> \rho </math> and the expectation value of an observable <math> A </math> by the scalar product <math> (\rho \cdot A) = \rm{tr}\{ \rho A \} </math>. There is no way to know if the combined system is pure from the knowledge of observables of the subsystem alone. In particular, if the combined system has [[quantum entanglement]], the state of the subsystem is not pure.