Naimark's Dilation Theorem - Finite Dimensional Case

Finite Dimensional Case

In the finite dimensional case, there is a somewhat more explicit formulation.

Suppose now, therefore C(X) is the finite dimensional algebra, and H has finite dimension m. A positive operator-valued measure E then assigns each i a positive semidefinite m X m matrix . Naimark's theorem now says there is a projection valued measure on X whose restriction is E.

Of particular interest is the special case when where I is the identity operator. (See the article on POVM for relevant applications.) This would mean the induced map is unital. It can be assumed with no loss of generality that each is a rank-one projection onto some . Under such assumptions, the case is excluded and we must have either:

1) and E is already a projection valued measure. (Because if and only if is an orthonormal basis.) ,or

2) and does not consist of mutually orthogonal projections.

For the second possibility, the problem of finding a suitable PVM now becomes the following: By assumption, the non-square matrix

is an isometry, i.e. . If we can find a matrix N where

is a n X n unitary matrix, the PVM whose elements are projections onto the column vectors of U will then have the desired properties. In principle, such a N can always be found.