Pure States and Measurements

Def Complete Family of Linear Maps A finite family of linear maps $\{f_i:H\to H:i\in I\}$ is complete when the following holds: ${\sum }_{i\in I}{f}_i={\mathrm{id}}_H$ Def
A family of linear maps H fi i ̸= j, the following holds:

H is orthogonal when for any (0.36)

Def Projection-Valued Measure A projection-valued measure (PVM) on a Hilbert space $H$ is a finite family of projections $\{p_i:H\to H:i\in I\}$ which are complete and orthogonal. A PVM is nondegenerate when Tr(pi) = 1 for all i.