Pure state
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The term pure state refers to several related concepts in physics, particularly quantum mechanics and in functional analysis. In quantum mechanics a pure state S of a quantum system is a state represented by a density operator which cannot be decomposed as a randomization of two statistically different statistical ensembles. Mathematically this means S is an extreme point in the set of states. Such states are given in Dirac bra-ket notation by
- <math> S = | \psi \rangle \langle \psi | <math>
In the density operator formalism, a pure state is an idempotent transformation, following the properties of projection operators
- <math> \rho = \rho^2 <math>
A pure state on a C*-algebra A is a state which is an extreme point of the set of all states on A. By properties of the GNS construction these states correspond to irreducible representations of A.
The states of the C*-algebra of compact operators K(H) correspond exactly to the density operators and therefore the pure states of K(H) are exactly the pure states in the sense of quantum mechanics.