Suppose that a random variable has a uniform distribution on a unit sphere.

Let be a collection of distinct points on the unit sphere centered at the origin.

A unit sphere is simply a sphere of radius one.

In this formalism, each state is represented as a point on the unit sphere.

By definition, the set of all such vectors forms the unit sphere.

All points of must lie on the unit sphere because .

The same applies for the radius if it is taken to equal one, as in the case of a unit sphere.

Let be the unit sphere in three dimensional Euclidean space.

Thus the set of all pure states corresponds to the unit sphere in the Hilbert space.

But this is the condition of being orthogonal to the unit sphere.