Naimark's dilation theorem

Summary

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.

Some preliminary notions

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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  , we have

 

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

  • E is called regular if the scalar valued measure
 

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.

  • E is called bounded if  .
  • E is called positive if E(B) is a positive operator for all B.
  • E is called self-adjoint if E(B) is self-adjoint for all B.
  • E is called spectral if it is self-adjoint and   for all  .

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:

  • If E is positive, then  , viewed as a map between C*-algebras, is also positive.
  •   is a homomorphism if, by definition, for all continuous f on X and  ,
 

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.

  • Similarly, to say   respects the * operation means
 

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.

  • Combining the previous two facts gives the conclusion that   is a *-homomorphism if and only if E is spectral and self adjoint. (When E is spectral and self adjoint, E is said to be a projection-valued measure or PVM.)

Naimark's theorem

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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

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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

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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   takes the form   for some potentially subnorrmalized vector  . 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 a co-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.

Spelling

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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.

References

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  • V. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, 2003.