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Michael J Kozdron - One of the best experts on this subject based on the ideXlab platform.

  • conditional expectation and bayes rule for quantum random variables and positive operator valued measures
    Journal of Mathematical Physics, 2012
    Co-Authors: Douglas Farenick, Michael J Kozdron
    Abstract:

    A quantum probability measure ν is a Function on a σ-algebra of subsets of a (locally compact and Hausdorff) sample space that satisfies the formal requirements for a measure, but where the values of ν are positive operators acting on a complex Hilbert space, and a quantum random variable is a measurable operator valued Function. Although quantum probability measures and random variables are used extensively in quantum mechanics, some of the fundamental probabilistic features of these structures remain to be determined. In this paper, we take a step toward a better mathematical understanding of quantum random variables and quantum probability measures by introducing a quantum analogue for the expected value Eν[ψ] of a quantum random variable ψ relative to a quantum probability measure ν. In so doing we are led to theorems for a change of quantum measure and a change of quantum variables. We also introduce a quantum conditional expectation which results in quantum versions of some standard identities for R...

  • conditional expectation and bayes rule for quantum random variables and positive operator valued measures
    arXiv: Probability, 2011
    Co-Authors: Douglas Farenick, Michael J Kozdron
    Abstract:

    A quantum probability measure is a Function on a sigma-algebra of subsets of a (locally compact and Hausdorff) sample space that satisfies the formal requirements for a measure, but whose values are positive operators acting on a complex Hilbert space, and a quantum random variable is a measurable operator valued Function. Although quantum probability measures and random variables are used extensively in quantum mechanics, some of the fundamental probabilistic features of these structures remain to be determined. In this paper we take a step toward a better mathematical understanding of quantum random variables and quantum probability measures by introducing a quantum analogue for the expected value of a quantum random variable relative to a quantum probability measure. In so doing we are led to theorems for a change of quantum measure and a change of quantum variables. We also introduce a quantum conditional expectation which results in quantum versions of some standard identities for Radon-Nikodym derivatives. This allows us to formulate and prove a quantum analogue of Bayes' rule.

Yaogan Mensah - One of the best experts on this subject based on the ideXlab platform.

Douglas Farenick - One of the best experts on this subject based on the ideXlab platform.

  • conditional expectation and bayes rule for quantum random variables and positive operator valued measures
    Journal of Mathematical Physics, 2012
    Co-Authors: Douglas Farenick, Michael J Kozdron
    Abstract:

    A quantum probability measure ν is a Function on a σ-algebra of subsets of a (locally compact and Hausdorff) sample space that satisfies the formal requirements for a measure, but where the values of ν are positive operators acting on a complex Hilbert space, and a quantum random variable is a measurable operator valued Function. Although quantum probability measures and random variables are used extensively in quantum mechanics, some of the fundamental probabilistic features of these structures remain to be determined. In this paper, we take a step toward a better mathematical understanding of quantum random variables and quantum probability measures by introducing a quantum analogue for the expected value Eν[ψ] of a quantum random variable ψ relative to a quantum probability measure ν. In so doing we are led to theorems for a change of quantum measure and a change of quantum variables. We also introduce a quantum conditional expectation which results in quantum versions of some standard identities for R...

  • conditional expectation and bayes rule for quantum random variables and positive operator valued measures
    arXiv: Probability, 2011
    Co-Authors: Douglas Farenick, Michael J Kozdron
    Abstract:

    A quantum probability measure is a Function on a sigma-algebra of subsets of a (locally compact and Hausdorff) sample space that satisfies the formal requirements for a measure, but whose values are positive operators acting on a complex Hilbert space, and a quantum random variable is a measurable operator valued Function. Although quantum probability measures and random variables are used extensively in quantum mechanics, some of the fundamental probabilistic features of these structures remain to be determined. In this paper we take a step toward a better mathematical understanding of quantum random variables and quantum probability measures by introducing a quantum analogue for the expected value of a quantum random variable relative to a quantum probability measure. In so doing we are led to theorems for a change of quantum measure and a change of quantum variables. We also introduce a quantum conditional expectation which results in quantum versions of some standard identities for Radon-Nikodym derivatives. This allows us to formulate and prove a quantum analogue of Bayes' rule.

Fritz Gesztesy - One of the best experts on this subject based on the ideXlab platform.

  • On Factorizations of Analytic Operator-Valued Functions and Eigenvalue Multiplicity Questions
    arXiv: Spectral Theory, 2014
    Co-Authors: Fritz Gesztesy, Helge Holden, Roger Nichols
    Abstract:

    We study several natural multiplicity questions that arise in the context of the Birman-Schwinger principle applied to non-self-adjoint operators. In particular, we re-prove (and extend) a recent result by Latushkin and Sukhtyaev by employing a different technique based on factorizations of analytic Operator-Valued Functions due to Howland. Factorizations of analytic Operator-Valued Functions are of particular interest in themselves and again we re-derive Howland's results and subsequently extend them. Considering algebraic multiplicities of finitely meromorphic Operator-Valued Functions, we recall the notion of the index of a finitely meromorphic Operator-Valued Function and use that to prove an analog of the well-known Weinstein-Aronszajn formula relating algebraic multiplicities of the underlying unperturbed and perturbed operators. Finally, we consider pairs of projections for which the difference belongs to the trace class and relate their Fredholm index to the index of the naturally underlying Birman-Schwinger operator.

  • Initial Value Problems and Weyl--Titchmarsh Theory for Schr\
    arXiv: Spectral Theory, 2011
    Co-Authors: Fritz Gesztesy, Rudi Weikard, Maxim Zinchenko
    Abstract:

    We develop Weyl-Titchmarsh theory for self-adjoint Schr\"odinger operators $H_{\alpha}$ in $L^2((a,b);dx;\cH)$ associated with the Operator-Valued differential expression $\tau =-(d^2/dx^2)+V(\cdot)$, with $V:(a,b)\to\cB(\cH)$, and $\cH$ a complex, separable Hilbert space. We assume regularity of the left endpoint $a$ and the limit point case at the right endpoint $b$. In addition, the bounded self-adjoint operator $\alpha= \alpha^* \in \cB(\cH)$ is used to parametrize the self-adjoint boundary condition at the left endpoint $a$ of the type $$ \sin(\alpha)u'(a)+\cos(\alpha)u(a)=0, $$ with $u$ lying in the domain of the underlying maximal operator $H_{\max}$ in $L^2((a,b);dx;\cH)$ associated with $\tau$. More precisely, we establish the existence of the Weyl-Titchmarsh solution of $H_{\alpha}$, the corresponding Weyl-Titchmarsh $m$-Function $m_{\alpha}$ and its Herglotz property, and determine the structure of the Green's Function of $H_{\alpha}$. Developing Weyl-Titchmarsh theory requires control over certain (Operator-Valued) solutions of appropriate initial value problems. Thus, we consider existence and uniqueness of solutions of 2nd-order differential equations with the operator coefficient $V$, -y" + (V - z) y = f \, \text{on} \, (a,b), y(x_0) = h_0, \; y'(x_0) = h_1, under the following general assumptions: $(a,b)\subseteq\bbR$ is a finite or infinite interval, $x_0\in(a,b)$, $z\in\bbC$, $V:(a,b)\to\cB(\cH)$ is a weakly measurable Operator-Valued Function with $\|V(\cdot)\|_{\cB(\cH)}\in L^1_\loc((a,b);dx)$, and $f\in L^1_{\loc}((a,b);dx;\cH)$, with $\cH$ a complex, separable Hilbert space. We also study the analog of this initial value problem with $y$ and $f$ replaced by Operator-Valued Functions $Y, F \in \cB(\cH)$. Our hypotheses on the local behavior of $V$ appear to be the most general ones to date.

  • initial value problems and weyl titchmarsh theory for schr odinger operators with operator valued potentials
    arXiv: Spectral Theory, 2011
    Co-Authors: Fritz Gesztesy, Rudi Weikard, Maxim Zinchenko
    Abstract:

    We develop Weyl-Titchmarsh theory for self-adjoint Schr\"odinger operators $H_{\alpha}$ in $L^2((a,b);dx;\cH)$ associated with the Operator-Valued differential expression $\tau =-(d^2/dx^2)+V(\cdot)$, with $V:(a,b)\to\cB(\cH)$, and $\cH$ a complex, separable Hilbert space. We assume regularity of the left endpoint $a$ and the limit point case at the right endpoint $b$. In addition, the bounded self-adjoint operator $\alpha= \alpha^* \in \cB(\cH)$ is used to parametrize the self-adjoint boundary condition at the left endpoint $a$ of the type $$ \sin(\alpha)u'(a)+\cos(\alpha)u(a)=0, $$ with $u$ lying in the domain of the underlying maximal operator $H_{\max}$ in $L^2((a,b);dx;\cH)$ associated with $\tau$. More precisely, we establish the existence of the Weyl-Titchmarsh solution of $H_{\alpha}$, the corresponding Weyl-Titchmarsh $m$-Function $m_{\alpha}$ and its Herglotz property, and determine the structure of the Green's Function of $H_{\alpha}$. Developing Weyl-Titchmarsh theory requires control over certain (Operator-Valued) solutions of appropriate initial value problems. Thus, we consider existence and uniqueness of solutions of 2nd-order differential equations with the operator coefficient $V$, -y" + (V - z) y = f \, \text{on} \, (a,b), y(x_0) = h_0, \; y'(x_0) = h_1, under the following general assumptions: $(a,b)\subseteq\bbR$ is a finite or infinite interval, $x_0\in(a,b)$, $z\in\bbC$, $V:(a,b)\to\cB(\cH)$ is a weakly measurable Operator-Valued Function with $\|V(\cdot)\|_{\cB(\cH)}\in L^1_\loc((a,b);dx)$, and $f\in L^1_{\loc}((a,b);dx;\cH)$, with $\cH$ a complex, separable Hilbert space. We also study the analog of this initial value problem with $y$ and $f$ replaced by Operator-Valued Functions $Y, F \in \cB(\cH)$. Our hypotheses on the local behavior of $V$ appear to be the most general ones to date.

Mawoussi Todjro - One of the best experts on this subject based on the ideXlab platform.