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Bulletin of the London Mathematical Society

Infinite sequences of linear functionals, positive operator-valued measures and Naimark extension theorem

Roberto Beneduci

Published Mar 17, 2010• 16 citations

Abstract

Let F : ℬ(ℝ)→ℱ(ℋ) be a commutative positive operator-valued measure from the Borel σ-algebra of the reals to the space of bounded positive operators on the Hilbert space ℋ. It is well known that there exist a self-adjoint operator A and a Markov kernel μ(·)(λ) such that F(Δ) = μΔ(A), where Δ ∈ ℬ(ℝ). We prove that, for any Δ ∈ ℬ(ℝ), if F(Δ) is a discrete operator, then A is a linear combination of the moments of F. This result allows us to characterize the quantum observable represented by A by means of the Naimark dilation of F. The result is connected with the following properties of the sequences of linear functionals. Consider an infinite sequence of linear functionals {Ti}i ∈ ℕ, such that Tif = ∫ f(t) dμt(i), corresponding to an infinite sequence of probability measures {μ(·)(i)}i ∈ ℕ on the Borel σ-algebra ℬ([0, 1]) such that μ(·)(i) ≠ μ(·)(j), where i, j ∈ ℕ and i≠j. There exists a real bounded one-to-one continuous function f such that T i f = ∫ f ( t ) d μ t ( i ) ≠ ∫ f ( t ) d μ t ( j ) = T j f , i , j ∊ ℕ , i ≠ j .

DOI: 10.1112/blms/bdq005

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Abstract After a short review of the main properties of unsharp observables we provide examples of commutative unsharp observables obtained as the projection of sharp observables. In particular, we consider a family of PVMs defined on the tensor product Hilbert space H ⊗ K which include the one introduced by Ozawa in his modification of the von Neumann position measurement model and show that their projections onto H are commutative POVMs. Then, we make some observations on the concept of Naimark dilation and its connections with the integral representation of commutative POVMs. Finally, we introduce a partial order relation in the set of Naimark dilations, show that there is a minimal element and that it is unique. Minimality is a consequence of the partial ordered structure and coincides with the usual definition of minimal Naimark dilation. In passing, we use Naimark’s theorem in order to prove some well known properties of positive operator valued measures.

Given a commutative POV measure [Formula: see text] on the Borel σ-algebra of the reals it is possible to construct a PV measure [Formula: see text] (the sharp reconstruction of F) such that the outcomes of the measurement of F can be interpreted as the random diffusion of the outcomes of the measurement of E. On the other hand Neumark's theorem ensures the existence of an extended Hilbert space [Formula: see text] and of a PV measure [Formula: see text] such that [Formula: see text] for every [Formula: see text]. We consider the projection [Formula: see text] of the self-adjoint operator [Formula: see text] corresponding to E+ and the self-adjoint operator A corresponding to the sharp reconstruction E and show the existence of a function G: [0,1] → [0,1] such that [Formula: see text]. An example for which the last relation can be inverted is shown.

Abstract In Kiukas, Lahti, and Ylinen (2006, Journal of Mathematical Physics 47, 072104), the authors asked the following general question. When is a positive operator measure projection valued? A version of this question formulated in terms of operator moments was posed in Pietrzycki and Stochel (2021, Journal of Functional Analysis 280, 109001). Let T be a self-adjoint operator, and let F be a Borel semispectral measure on the real line with compact support. For which positive integers

p< q

do the equalities

T^k =\int _{\mathbb {R}} x^k F(\mathrm {d\hspace {.1ex}} x)

,

k=p, q

, imply that F is a spectral measure? In the present paper, we completely solve the second problem. The answer is affirmative if

p

is odd and

q

is even, and negative otherwise. The case

(p,q)=(1,2)

closely related to intrinsic noise operator was solved by several authors including Kruszyński and de Muynck, as well as Kiukas, Lahti, and Ylinen. The counterpart of the second problem concerning the multiplicativity of unital positive linear maps on

C^*

-algebras is also provided.

Denoted by M(A), QM(A) and SQM(A) the sets of all measures, quantum measures and subadditive quantum measures on a σ-algebra A, respectively. We observe that these sets are all positive cones in the real vector space F(A) of all real-valued functions on A and prove that M(A) is a face of SQM(A). It is proved that the product of m grade-1 measures is a grade-m measure. By combining a matrix M µ to a quantum measure µ on the power set A n of an n-element set X, it is proved that µ ≪ ν (resp. µ ⊥ ν) if and only if M µ ≪ M ν (resp. M µ M ν =0). Also, it is shown that two nontrivial measures µ and ν are mutually absolutely continuous if and only if µ·ν∈QM(A n ). Moreover, the matrices corresponding to quantum measures are characterized. Finally, convergence of a sequence of quantum measures on A n is introduced and discussed; especially, the Vitali-Hahn-Saks theorem for quantum measures is proved.