structural:summarization
Density matrices

Wave functions are not always the most convenient way to describe quantum systems and their behavior. When the preparation of a system is only imperfectly known, or when the system under investigation is a part of a larger whole, density matrices may be used instead. A density matrix is a positive semi-definite operator whose trace is equal to 1. (The term "density operator" is also used, particularly when the underlying Hilbert space is infinite-dimensional.) The set of all density matrices is convex, and the extreme points are the operators that project onto vectors in the Hilbert space. These are the density-matrix representations of wave functions; in Dirac notation, they are written ρ ^ = | Ψ ⟩ ⟨ Ψ |. {\displaystyle {\hat {\rho }}=|\Psi \rangle \langle \Psi |.}

The density-matrix analogue of the Schrödinger equation for wave functions is i ℏ ∂ ρ ^ ∂ t =, {\displaystyle i\hbar {\frac {\partial {\hat {\rho }}}{\partial t}}=,} where the brackets denote a commutator. This is variously known as the von Neumann equation, the Liouville–von Neumann equation, or just the Schrödinger equation for density matrices. If the Hamiltonian is time-independent, this equation can be easily solved to yield ρ ^ (t) = e − i H ^ t / ℏ ρ ^ (0) e i H ^ t / ℏ. {\displaystyle {\hat {\rho }}(t)=e^{-i{\hat {H}}t/\hbar }{\hat {\rho }}(0)e^{i{\hat {H}}t/\hbar }.}

More generally, if the unitary operator U ^ (t) {\displaystyle {\hat {U}}(t)} describes wave function evolution over some time interval, then the time evolution of a density matrix over that same interval is given by ρ ^ (t) = U ^ (t) ρ ^ (0) U ^ (t) †. {\displaystyle {\hat {\rho }}(t)={\hat {U}}(t){\hat {\rho }}(0){\hat {U}}(t)^{\dagger }.}

Unitary evolution of a density matrix conserves its von Neumann entropy.