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Journal of Mathematical Physics

A geometrical characterization of commutative positive operator valued measures

Roberto Beneduci

Published Jun 1, 2006• 24 citations
Publisher

Abstract

We show that a POV measure F on the Borel σ-algebra of the reals B(R) is commutative if and only if there exists a PV measure E on B(R) and, for every λ in the spectrum of E, a probability measure γ(∙)(λ) on B(R) such that the effect F(Δ) coincides with γΔ(A), where A is the self-adjoint operator associated to E. The relevance of this result to the theory of the sharp reconstruction is analyzed.

DOI
10.1063/1.2206879

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The present dissertation focuses on the analysis of the mathematical structure of the Positive Operator Valued Measures (POVM) and their relevance to quantum mechanics. In particular we analyze: 1. The relationships between POVMs and PVMs (Projection Valued Measures) and prove that each commutative POVM F is the smearing (realized by a Feller Markov Kernel) of a spectral measure. That suggests an interpretation of commutative POVMs as the randomization of real PVMs. Moreover, we characterize the POVMs whose smearing can be realized by strong Feller Markov kernels. 2. The relationships between the characterization of commutative POVMs in item 1) and Naimark's dilation theorem. We prove that the self-adjoint operator A corresponding to the spectral measure E, of which F is the smearing, is the projection of a Naimark operator. 3. Analysis of the informational content of a POVM. We introduce an equivalence relation on the set of observables which we compare with other well known equivalence relations and prove that it is the only one for which E is always equivalent to F. 4. The uniform continuity of a POVM and its relevance to the problem of localization. We take into consideration a non-commutative POVM defined on a locally compact second countable Haussdorf topological space and give necessary and sufficient conditions for it to be uniformly continuous. Moreover, we show the relevance of this result to relativistic quantum mechanics.

•Jan 1, 2014•0 citations
We show that a POV measure F on the Borel σ-algebra of the reals B(R) is commutative if and only if there exists a PV measure E on B(R) and, for every λ in the spectrum of E, a probability measure γ(∙)(λ) on B(R) such that the effect F(Δ) coincides with γΔ(A), where A is the self-adjoint operator associated to E. The relevance of this result to the theory of the sharp reconstruction is analyzed.
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.
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.
We study two relevant characterizations of a commutative positive operator valued measure (POVM) F . The first one is a Choquet type of an integral representation. It introduces a measure ν on the space of the projection valued measures (PVMs) and describes F as an integral over this space. The second one represents a commutative POVM F as the randomization of a single PVM E by means of a Markov kernel μ . We show that one can be derived from the other. We also elaborate upon some previous results on Choquet’s representation of Markov kernels and find a functional relationship between ν and μ . Finally, we analyze some relevant particular cases and provide some physically relevant examples which include the unsharp position observables.
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 .
We give a characterization of commutative semispectral measures by means of Feller and Strong Feller Markov kernels. In particular: {itemize} we show that a semispectral measure FFF is commutative if and only if there exist a self-adjoint operator AAA and a Markov kernel μ(⋅)(⋅):Γ×B(R)→[0,1]μ_{(\cdot)}(\cdot):Γ\times\mathcal{B}(\mathbb{R})\to[0,1]μ(⋅)​(⋅):Γ×B(R)→[0,1], Γ⊂σ(A)Γ\subsetσ(A)Γ⊂σ(A), E(Γ)=1E(Γ)=\mathbf{1}E(Γ)=1, such that F(Δ)=∫ΓμΔ(λ) dEλ,F(Δ)=\int_Γμ_Δ(λ)\,dE_λ,F(Δ)=∫Γ​μΔ​(λ)dEλ​, \noindent and μ(Δ)μ_{(Δ)}μ(Δ)​ is continuous for each Δ∈RΔ\in RΔ∈R where, R⊂B(R)R\subset\mathcal{B}(\mathbb{R})R⊂B(R) is a ring which generates the Borel σσσ-algebra of the reals B(R)\mathcal{B}(\mathbb{R})B(R). Moreover, μ(⋅)(⋅)μ_{(\cdot)}(\cdot)μ(⋅)​(⋅) is a Feller Markov kernel and separates the points of ΓΓΓ. we prove that FFF admits a strong Feller Markov kernel μ(⋅)(⋅)μ_{(\cdot)}(\cdot)μ(⋅)​(⋅), if and only if FFF is uniformly continuous. Finally, we prove that if FFF is absolutely continuous with respect to a regular finite measure ννν then, it admits a strong Feller Markov kernel. {itemize} The mathematical and physical relevance of the results is discussed giving a particular emphasis to the connections between μμμ and the imprecision of the measurement apparatus.
The present dissertation focuses on the analysis of the mathematical structure of the Positive Operator Valued Measures (POVM) and their relevance to quantum mechanics. In particular we analyze: 1. The relationships between POVMs and PVMs (Projection Valued Measures) and prove that each commutative POVM F is the smearing (realized by a Feller Markov Kernel) of a spectral measure. That suggests an interpretation of commutative POVMs as the randomization of real PVMs. Moreover, we characterize the POVMs whose smearing can be realized by strong Feller Markov kernels. 2. The relationships between the characterization of commutative POVMs in item 1) and Naimark's dilation theorem. We prove that the self-adjoint operator A corresponding to the spectral measure E, of which F is the smearing, is the projection of a Naimark operator. 3. Analysis of the informational content of a POVM. We introduce an equivalence relation on the set of observables which we compare with other well known equivalence relations and prove that it is the only one for which E is always equivalent to F. 4. The uniform continuity of a POVM and its relevance to the problem of localization. We take into consideration a non-commutative POVM defined on a locally compact second countable Haussdorf topological space and give necessary and sufficient conditions for it to be uniformly continuous. Moreover, we show the relevance of this result to relativistic quantum mechanics.