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International Journal of Geometric Methods in Modern Physics

NEUMARK'S OPERATORS AND SHARP RECONSTRUCTIONS

Roberto Beneduci

Published Dec 1, 2006• 15 citations
Publisher

Abstract

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.

DOI
10.1142/s0219887806001831

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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.
A commutative positive operator valued (POV) measure F with real spectrum is characterized by the existence of a projection valued measure E (the sharp reconstruction of F) with real spectrum such that F can be interpreted as a randomization of E. This paper focuses on the relationships between this characterization of commutative POV measures and Neumark’s extension theorem. In particular, we show that in the finite dimensional case there exists a relation between the Neumark operator corresponding to the extension of F and the sharp reconstruction of F. The relevance of this result to the theory of nonideal quantum measurement and to the definition of unsharpness is analyzed.
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 .
Abstract The existence of a real linear space structure on the set of observables of a quantum system—i.e., the requirement that the linear combination of two generally non-commuting observables A, B is an observable as well—is a fundamental postulate of the quantum theory yet before introducing any structure of algebra. However, it is by no means clear how to choose the measuring instrument of a general observable of the form aA+bBaA+bBaA+bB a A + b B ( a,b∈Ra,b\in {{\mathbb {R}}}a,b∈R a , b ∈ R ) if such measuring instruments are given for the addends observables A and B when they are incompatible observables. A mathematical version of this dilemma is how to construct the spectral measure of f(aA+bB)f(aA+bB)f(aA+bB) f ( a A + b B ) out of the spectral measures of A and B. We present such a construction with a formula which is valid for general unbounded self-adjoint operators A and B, whose spectral measures may not commute, and a wide class of functions f:R→Cf: {{\mathbb {R}}}\rightarrow {{\mathbb {C}}}f:R→C f : R → C . In the bounded case, we prove that the Jordan product of A and B (and suitably symmetrized polynomials of A and B) can be constructed with the same procedure out of the spectral measures of A and B. The formula turns out to have an interesting operational interpretation and, in particular cases, a nice interplay with the theory of Feynman path integration and the Feynman–Kac formula.