Naimark's dilation theorem

In operator theory, Naimark's dilation theorem is a result that characterizes positive operator valued measures. It can be viewed as a consequence of Stinespring's dilation theorem.

Note

In the mathematical literature, one may also find other results that bear Naimark's name.

Spelling

In the physics literature, it is common to see the spelling “Neumark” instead of “Naimark.” The latter variant is according to the romanization of Russian used in translation of Soviet journals, with diacritics omitted (originally Naĭmark). The former is according to the etymology of the surname.

Some preliminary notions

Let X be a compact Hausdorff space, H be a Hilbert space, and L(H) the Banach space of bounded operators on H. A mapping E from the Borel σ-algebra on X to is called a operator-valued measure if it is weakly countably additive, that is, for any disjoint sequence of Borel sets , we have

for all x and y. Some terminology for describing such measures are:

is a regular Borel measure, meaning all compact sets have finite total variation and the measure of a set can be approximated by those of open sets.

We will assume throughout that E is regular.

Let C(X) denote the abelian C*-algebra of continuous functions on X. If E is regular and bounded, it induces a map in the obvious way:

The boundedness of E implies, for all h of unit norm

This shows is a bounded operator for all f, and itself is a bounded linear map as well.

The properties of are directly related to those of E:

Take f and g to be indicator functions of Borel sets and we see that is a homomorphism if and only if E is spectral.

The LHS is

and the RHS is

So, taking f a sequence of continuous functions increasing to the indicator function of B, we get , i.e. E(B) is self adjoint.

Naimark's theorem

The theorem reads as follows: Let E be a positive L(H)-valued measure on X. There exists a Hilbert space K, a bounded operator , and a self-adjoint, spectral L(K)-valued measure on X, F, such that

Proof

We now sketch the proof. The argument passes E to the induced map and uses Stinespring's dilation theorem. Since E is positive, so is as a map between C*-algebras, as explained above. Furthermore, because the domain of , C(X), is an abelian C*-algebra, we have that is completely positive. By Stinespring's result, there exists a Hilbert space K, a *-homomorphism , and operator such that

Since π is a *-homomorphism, its corresponding operator-valued measure F is spectral and self adjoint. It is easily seen that F has the desired properties.

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 × m matrix . Naimark's theorem now states that 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.) In this case, 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),
  2. and does not consist of mutually orthogonal projections.

For the second possibility, the problem of finding a suitable projection-valued measure now becomes the following problem. By assumption, the non-square matrix

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

is a n × n unitary matrix, the projection-valued measure 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.

References

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