Editing Positive operator (section)

== Application to physics: quantum states ==
{{Main|Quantum state|Density operator}}
The definition of a [[quantum system]] includes a complex [[separable Hilbert space]] <math>H_\mathbb{C}</math> and a set <math>\cal S</math> of positive [[trace-class]] [[density operator|operators]] <math>\rho</math> on <math>H_\mathbb{C}</math> for which <math>\mathop{\text{Trace}}\rho = 1.</math> The [[Set (mathematics)|set]] <math>\cal S</math> is ''the set of states''. Every <math>\rho \in {\cal S}</math> is called a ''state'' or a ''density operator''. For <math>\psi \in H_\mathbb{C},</math> where <math>\|\psi\| = 1,</math> the operator <math>P_\psi</math> of projection onto the [[Linear span|span]] of <math>\psi</math> is called a ''[[pure state]]''. (Since each pure state is identifiable with a [[unit vector]] <math>\psi \in H_\mathbb{C},</math> some sources define pure states to be unit elements from <math>H_\mathbb{C}).</math> States that are not pure are called ''[[Mixed state (physics)|mixed]]''.