Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2103.09964 · Canadian Journal of Mathematics

Two-moment characterization of spectral measures on the real line

Paweł Pietrzycki, Jan Stochel

Published Sep 15, 2022• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.4153/s0008414x22000426

Related papers

Phase space quantization as a moment problem

Jukka Kiukas

We consider questions related to the following quantization scheme: a classical variable f: Ω → ℝ on a phase space Ω is associated with a unique semispectral measure E f , such that the kth moment operator of E f is required to coincide with the operator integral L(f k , E) of f k with respect to a certain fixed phase space semispectral measure E. Mainly, we take the phase space Ω to be a locally compact unimodular group. In the concrete case where Ω = ℝ2 and E is a translation covariant semispectral measure, we determine explicitly the relevant operators L(f k , E) for certain variables f. In addition, we consider the question under what conditions a positive operator measure is projection valued.

Optics and Spectroscopy•Sep 1, 2007•1 citation

Phase space quantization and the operator moment problem

Jukka Kiukas, PEKKA J. LAHTI, Kari Ylinen

We consider questions related to a quantization scheme in which a classical variable f:Ω→R on a phase space Ω is associated with a (preferably unique) semispectral measure Ef, such that the moment operators of Ef are required to be of the form Γ(fk), with Γ a suitable mapping from the set of classical variables to the set of (not necessarily bounded) operators in the Hilbert space of the quantum system. In particular, we investigate the situation where the map Γ is implemented by the operator integral with respect to some fixed positive operator measure. The phase space Ω is first taken to be an abstract measurable space, then a locally compact unimodular group, and finally R2, where we determine explicitly the relevant operators Γ(fk) for certain variables f, in the case where the quantization map Γ is implemented by a translation covariant positive operator measure. In addition, we consider the question under what conditions a positive operator measure is projection valued.

Journal of Mathematical Physics•Jul 1, 2006•23 citations

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

Roberto Beneduci

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 .

Bulletin of the London Mathematical Society•Mar 17, 2010•16 citations

Semispectral Measures and Feller markov Kernels

Roberto Beneduci

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 $F$ is commutative if and only if there exist a self-adjoint operator $A$ and a Markov kernel $μ_{(\cdot)}(\cdot):Γ\times\mathcal{B}(\mathbb{R})\to[0,1]$, $Γ\subsetσ(A)$, $E(Γ)=\mathbf{1}$, such that $$F(Δ)=\int_Γμ_Δ(λ)\,dE_λ,$$ \noindent and $μ_{(Δ)}$ is continuous for each $Δ\in R$ where, $R\subset\mathcal{B}(\mathbb{R})$ is a ring which generates the Borel $σ$-algebra of the reals $\mathcal{B}(\mathbb{R})$. Moreover, $μ_{(\cdot)}(\cdot)$ is a Feller Markov kernel and separates the points of $Γ$. we prove that $F$ admits a strong Feller Markov kernel $μ_{(\cdot)}(\cdot)$, if and only if $F$ is uniformly continuous. Finally, we prove that if $F$ is absolutely continuous with respect to a regular finite measure $ν$ 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 $μ$ and the imprecision of the measurement apparatus.

arXiv:1207.0086•Jun 30, 2012•3 citations

Notes on phase space quantization

Jukka Kiukas, PEKKA J. LAHTI, Kari Ylinen

We consider questions related to a quantization scheme in which a classical variable f:Ω\to R on a phase space Ωis associated with a semispectral measure E^f, such that the moment operators of E^f are required to be of the form Γ(f^k), with Γa suitable mapping from the set of classical variables to the set of (not necessarily bounded) operators in some Hilbert space. In particular, we investigate the situation where the map Γis implemented by the operator integral with respect to some fixed positive operator measure. The phase space Ωis first taken to be an abstract measurable space, then a locally compact unimodular group, and finally R^2, where we determine explicitly the relevant operators Γ(f^k) for certain variables f, in the case where the quantization map Γis implemented by a translation covariant positive operator measure. In addition, we consider the question under what conditions a positive operator measure is projection valued.

arXiv:quant-ph/0601009•Jan 2, 2006•0 citations

A geometrical characterization of commutative positive operator valued measures

Roberto Beneduci

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.

Journal of Mathematical Physics•Jun 1, 2006•24 citations
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<qp< qp<q do the equalities Tk=∫RxkF(dx)T^k =\int _{\mathbb {R}} x^k F(\mathrm {d\hspace {.1ex}} x)Tk=∫R​xkF(dx) , k=p,qk=p, qk=p,q , imply that F is a spectral measure? In the present paper, we completely solve the second problem. The answer is affirmative if ppp is odd and qqq is even, and negative otherwise. The case (p,q)=(1,2)(p,q)=(1,2)(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∗C^*C∗ -algebras is also provided.
We consider questions related to the following quantization scheme: a classical variable f: Ω → ℝ on a phase space Ω is associated with a unique semispectral measure E f , such that the kth moment operator of E f is required to coincide with the operator integral L(f k , E) of f k with respect to a certain fixed phase space semispectral measure E. Mainly, we take the phase space Ω to be a locally compact unimodular group. In the concrete case where Ω = ℝ2 and E is a translation covariant semispectral measure, we determine explicitly the relevant operators L(f k , E) for certain variables f. In addition, we consider the question under what conditions a positive operator measure is projection valued.
We consider questions related to a quantization scheme in which a classical variable f:Ω→R on a phase space Ω is associated with a (preferably unique) semispectral measure Ef, such that the moment operators of Ef are required to be of the form Γ(fk), with Γ a suitable mapping from the set of classical variables to the set of (not necessarily bounded) operators in the Hilbert space of the quantum system. In particular, we investigate the situation where the map Γ is implemented by the operator integral with respect to some fixed positive operator measure. The phase space Ω is first taken to be an abstract measurable space, then a locally compact unimodular group, and finally R2, where we determine explicitly the relevant operators Γ(fk) for certain variables f, in the case where the quantization map Γ is implemented by a translation covariant positive operator measure. In addition, we consider the question under what conditions a positive operator measure is projection valued.
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.
We consider questions related to a quantization scheme in which a classical variable f:Ω\to R on a phase space Ωis associated with a semispectral measure E^f, such that the moment operators of E^f are required to be of the form Γ(f^k), with Γa suitable mapping from the set of classical variables to the set of (not necessarily bounded) operators in some Hilbert space. In particular, we investigate the situation where the map Γis implemented by the operator integral with respect to some fixed positive operator measure. The phase space Ωis first taken to be an abstract measurable space, then a locally compact unimodular group, and finally R^2, where we determine explicitly the relevant operators Γ(f^k) for certain variables f, in the case where the quantization map Γis implemented by a translation covariant positive operator measure. In addition, we consider the question under what conditions a positive operator measure is projection valued.