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.
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 an operator-valued measure if it is weakly countably additive, that is, for any disjoint sequence of Borel sets
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.
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
are directly related to those of E: Take f and g to be indicator functions of Borel sets and we see that
The LHS is and the RHS is So, taking f a sequence of continuous functions increasing to the indicator function of B, we get
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 F on X, such that We now sketch the proof.
The argument passes E to the induced map
By Stinespring's result, there exists a Hilbert space K, a *-homomorphism
such that Since π is a *-homomorphism, its corresponding operator-valued measure F is spectral and self adjoint.
In the finite-dimensional case, there is a somewhat more explicit formulation.
, 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
(See the article on POVM for relevant applications.)
In this case, the induced map
for some potentially subnorrmalized vector
is excluded and we must have either 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 a co-isometry, that is
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 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 of Mark Naimark.