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.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, 20140 citations

Mathematical structure of positive operator valued measures and applications

Roberto Beneduci

Nov 17, 20143 citations

Positive Operator-Valued Measures and Frames

S. Twareque Ali, Jean-Pierre Antoine, Jean‐Pierre Gazeau

Theoretical and mathematical physicsAug 26, 20131 citation

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.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:1207.0086Jun 30, 20123 citations

Positive Operator-Valued Measures and Frames

S. Twareque Ali, Jean-Pierre Antoine, Jean‐Pierre Gazeau

Graduate texts in contemporary physicsJan 1, 20000 citations

Composing Quantum Instruments

Robert I. Booth, Dominik Leichtle, Alex Rice +1 more

We study the composition of classically-controlled quantum instruments--the natural quantum analogue of Markov kernels. Classically, Markov kernels compose by integrating one kernel against another. Defining this composition for quantum instruments with continuous outcomes requires an integral of quantum channel-valued functions with respect to a quantum instrument. We construct this integral in the Heisenberg picture using the Okamura-Ozawa normal extension to a von Neumann tensor product. This integral recovers the expected finite formula, preserves normal complete positivity and subunitality, and provides the multiplication for a monad governing the composition of quantum instruments. As an immediate consequence, we identify the category of quantum Markov kernels as the Kleisli category of this monad.We study the composition of classically-controlled quantum instruments--the natural quantum analogue of Markov kernels. Classically, Markov kernels compose by integrating one kernel against another. Defining this composition for quantum instruments with continuous outcomes requires an integral of quantum channel-valued functions with respect to a quantum instrument. We construct this integral in the Heisenberg picture using the Okamura-Ozawa normal extension to a von Neumann tensor product. This integral recovers the expected finite formula, preserves normal complete positivity and subunitality, and provides the multiplication for a monad governing the composition of quantum instruments. As an immediate consequence, we identify the category of quantum Markov kernels as the Kleisli category of this monad.

quant-pharXiv:2606.28291Jun 29, 20260 citations

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.