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1. Pure states as projection operators

For a spin $1/2$ quantum system pure quantum states are uniquely described by projection operators $P({\bf n})$, where ${\bf n}$ is a unit length direction vector, starting at the origin, and ending on one of the points of the unit sphere. All pure states are of this form. Indeed, every orthogonal projection, except of the two trivial ones: $0$ and $I,$ are of the form $P({\bf n})$ for some ${\bf n}.$