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Hendra I. Nurdin - One of the best experts on this subject based on the ideXlab platform.

  • error bounds for Finite Dimensional approximations of input output open quantum systems by Subspace truncation and adiabatic elimination
    arXiv: Quantum Physics, 2015
    Co-Authors: Onvaree Techakesari, Hendra I. Nurdin
    Abstract:

    An important class of physical systems that are of interest in practice are input-output open quantum systems that can be described by quantum stochastic differential equations and defined on an inFinite-Dimensional underlying Hilbert space. Most commonly, these systems involve coupling to a quantum harmonic oscillator as a system component. This paper is concerned with error bounds in the Finite-Dimensional approximations of input-output open quantum systems defined on an inFinite-Dimensional Hilbert space. We develop a framework for developing error bounds between the time evolution of the state of a class of inFinite-Dimensional quantum systems and its approximation on a Finite-Dimensional Subspace of the original, when both are initialized in the latter Subspace. This framework is then applied to two approaches for obtaining Finite-Dimensional approximations: Subspace truncation and adiabatic elimination. Applications of the bounds to some physical examples drawn from the literature are provided to illustrate our results.

  • Error bounds on Finite-Dimensional approximations of input-output open quantum systems
    2015 54th IEEE Conference on Decision and Control (CDC), 2015
    Co-Authors: Onvaree Techakesari, Hendra I. Nurdin
    Abstract:

    Many physical systems of interest that are encountered in practice are input-output open quantum systems described by quantum stochastic differential equations and defined on an inFinite-Dimensional underlying Hilbert space. Most commonly, these systems involve coupling to a quantum harmonic oscillator as a system component. This paper is concerned with the error in the Finite-Dimensional approximation of input-output open quantum systems defined on an inFinite-Dimensional underlying Hilbert space. We present explicit error bounds between the time evolution of the state of a class of inFinite-Dimensional quantum systems and its approximation on a Finite-Dimensional Subspace of the original, when both are initialized in the latter Subspace. Application to a physical example drawn from the literature is provided to illustrate our results.

Martin Ziegler - One of the best experts on this subject based on the ideXlab platform.

  • Definable relations in Finite Dimensional Subspace lattices with involution. Part II: Quantifier-free and homogeneous descriptions
    Algebra universalis, 2019
    Co-Authors: Christian Herrmann, Martin Ziegler
    Abstract:

    For Finite Dimensional hermitean inner product spaces V , over $$*$$ ∗ -fields F , and in the presence of orthogonal bases providing form elements in the prime subfield of F , we show that quantifier-free definable relations in the Subspace lattice $$\mathsf{L}(V)$$ L ( V ) , endowed with the involution induced by orthogonality, admit quantifier-free descriptions within F , also in terms of Grassmann–Plücker coordinates. In the latter setting, homogeneous descriptions are obtained if one allows quantification type $$\Sigma _1$$ Σ 1 . In absence of involution, these results remain valid.

  • quantifier free definable relations on Finite Dimensional Subspace lattices with involution
    arXiv: Logic, 2018
    Co-Authors: Christian Herrmann, Martin Ziegler
    Abstract:

    For Finite Dimensional hermitean inner product spaces $V$, over $*$-fields $F$, and in the presence of orthogonal bases providing form elements in the prime subfield of $F$, we show that quantifier free definable relations in the Subspace lattice $L(V)$ with involution by taking orthogonals, admit quantifier free descriptions within $F$, also in terms of Grassmann-Plucker coordinates.

  • definable relations in Finite Dimensional Subspace lattices with involution part ii quantifier free and homogeneous descriptions
    arXiv: Logic, 2018
    Co-Authors: Christian Herrmann, Martin Ziegler
    Abstract:

    For Finite Dimensional hermitean inner product spaces $V$, over $*$-fields $F$, and in the presence of orthogonal bases providing form elements in the prime subfield of $F$, we show that quantifier free definable relations in the Subspace lattice $L(V)$ with involution by taking orthogonals, admit quantifier free descriptions within $F$, also in terms of Grassmann-Pl\"ucker coordinates.In the latter setting, homogeneous descriptions are obtained if one allows quantification type $\Sigma_1$. In absence of involution, these results remain valid.

  • definable relations in Finite Dimensional Subspace lattices with involution
    Algebra Universalis, 2018
    Co-Authors: Christian Herrmann, Martin Ziegler
    Abstract:

    For a large class of Finite Dimensional inner product spaces V, over division \(*\)-rings F, we consider definable relations on the Subspace lattice \(\mathsf{L}(V)\) of V, endowed with the operation of taking orthogonals. In particular, we establish translations between the relevant first order languages, in order to associate these relations with definable and invariant relations on F—focussing on the quantification type of defining formulas. As an intermediate structure we consider the \(*\)-ring \(\mathsf{R}(V)\) of endomorphisms of V, thereby identifying \(\mathsf{L}(V)\) with the lattice of right ideals of \(\mathsf{R}(V)\), with the induced involution. As an application, model completeness of F is shown to imply that of \(\mathsf{R}(V)\) and \(\mathsf{L}(V)\).

Onvaree Techakesari - One of the best experts on this subject based on the ideXlab platform.

  • error bounds for Finite Dimensional approximations of input output open quantum systems by Subspace truncation and adiabatic elimination
    arXiv: Quantum Physics, 2015
    Co-Authors: Onvaree Techakesari, Hendra I. Nurdin
    Abstract:

    An important class of physical systems that are of interest in practice are input-output open quantum systems that can be described by quantum stochastic differential equations and defined on an inFinite-Dimensional underlying Hilbert space. Most commonly, these systems involve coupling to a quantum harmonic oscillator as a system component. This paper is concerned with error bounds in the Finite-Dimensional approximations of input-output open quantum systems defined on an inFinite-Dimensional Hilbert space. We develop a framework for developing error bounds between the time evolution of the state of a class of inFinite-Dimensional quantum systems and its approximation on a Finite-Dimensional Subspace of the original, when both are initialized in the latter Subspace. This framework is then applied to two approaches for obtaining Finite-Dimensional approximations: Subspace truncation and adiabatic elimination. Applications of the bounds to some physical examples drawn from the literature are provided to illustrate our results.

  • Error bounds on Finite-Dimensional approximations of input-output open quantum systems
    2015 54th IEEE Conference on Decision and Control (CDC), 2015
    Co-Authors: Onvaree Techakesari, Hendra I. Nurdin
    Abstract:

    Many physical systems of interest that are encountered in practice are input-output open quantum systems described by quantum stochastic differential equations and defined on an inFinite-Dimensional underlying Hilbert space. Most commonly, these systems involve coupling to a quantum harmonic oscillator as a system component. This paper is concerned with the error in the Finite-Dimensional approximation of input-output open quantum systems defined on an inFinite-Dimensional underlying Hilbert space. We present explicit error bounds between the time evolution of the state of a class of inFinite-Dimensional quantum systems and its approximation on a Finite-Dimensional Subspace of the original, when both are initialized in the latter Subspace. Application to a physical example drawn from the literature is provided to illustrate our results.

Christian Herrmann - One of the best experts on this subject based on the ideXlab platform.

  • Definable relations in Finite Dimensional Subspace lattices with involution. Part II: Quantifier-free and homogeneous descriptions
    Algebra universalis, 2019
    Co-Authors: Christian Herrmann, Martin Ziegler
    Abstract:

    For Finite Dimensional hermitean inner product spaces V , over $$*$$ ∗ -fields F , and in the presence of orthogonal bases providing form elements in the prime subfield of F , we show that quantifier-free definable relations in the Subspace lattice $$\mathsf{L}(V)$$ L ( V ) , endowed with the involution induced by orthogonality, admit quantifier-free descriptions within F , also in terms of Grassmann–Plücker coordinates. In the latter setting, homogeneous descriptions are obtained if one allows quantification type $$\Sigma _1$$ Σ 1 . In absence of involution, these results remain valid.

  • quantifier free definable relations on Finite Dimensional Subspace lattices with involution
    arXiv: Logic, 2018
    Co-Authors: Christian Herrmann, Martin Ziegler
    Abstract:

    For Finite Dimensional hermitean inner product spaces $V$, over $*$-fields $F$, and in the presence of orthogonal bases providing form elements in the prime subfield of $F$, we show that quantifier free definable relations in the Subspace lattice $L(V)$ with involution by taking orthogonals, admit quantifier free descriptions within $F$, also in terms of Grassmann-Plucker coordinates.

  • definable relations in Finite Dimensional Subspace lattices with involution part ii quantifier free and homogeneous descriptions
    arXiv: Logic, 2018
    Co-Authors: Christian Herrmann, Martin Ziegler
    Abstract:

    For Finite Dimensional hermitean inner product spaces $V$, over $*$-fields $F$, and in the presence of orthogonal bases providing form elements in the prime subfield of $F$, we show that quantifier free definable relations in the Subspace lattice $L(V)$ with involution by taking orthogonals, admit quantifier free descriptions within $F$, also in terms of Grassmann-Pl\"ucker coordinates.In the latter setting, homogeneous descriptions are obtained if one allows quantification type $\Sigma_1$. In absence of involution, these results remain valid.

  • definable relations in Finite Dimensional Subspace lattices with involution
    Algebra Universalis, 2018
    Co-Authors: Christian Herrmann, Martin Ziegler
    Abstract:

    For a large class of Finite Dimensional inner product spaces V, over division \(*\)-rings F, we consider definable relations on the Subspace lattice \(\mathsf{L}(V)\) of V, endowed with the operation of taking orthogonals. In particular, we establish translations between the relevant first order languages, in order to associate these relations with definable and invariant relations on F—focussing on the quantification type of defining formulas. As an intermediate structure we consider the \(*\)-ring \(\mathsf{R}(V)\) of endomorphisms of V, thereby identifying \(\mathsf{L}(V)\) with the lattice of right ideals of \(\mathsf{R}(V)\), with the induced involution. As an application, model completeness of F is shown to imply that of \(\mathsf{R}(V)\) and \(\mathsf{L}(V)\).

Ursula Molter - One of the best experts on this subject based on the ideXlab platform.

  • REFINABLE SHIFT INVARIANT SPACES IN ℝd
    International Journal of Wavelets Multiresolution and Information Processing, 2020
    Co-Authors: Carlos Cabrelli, Sigrid B. Heineken, Ursula Molter
    Abstract:

    Let φ : ℝd → ℂ be a compactly supported function which satisfies a refinement equation of the form [Formula: see text] where Γ ⊂ ℝd is a lattice, Λ is a Finite subset of Γ, and A is a dilation matrix. We prove, under the hypothesis of linear independence of the Γ-translates of φ, that there exists a correspondence between the vectors of the Jordan basis of a Finite submatrix of L = [cAi-j]i,j∈Γ and a Finite-Dimensional Subspace [Formula: see text] in the shift-invariant space generated by φ. We provide a basis of [Formula: see text] and show that its elements satisfy a property of homogeneity associated to the eigenvalues of L. If the function φ has accuracy κ, this basis can be chosen to contain a basis for all the multivariate polynomials of degree less than κ. These latter functions are associated to eigenvalues that are powers of the eigenvalues of A-1. Furthermore we show that the dimension of [Formula: see text] coincides with the local dimension of φ, and hence, every function in the shift-invariant space generated by φ can be written locally as a linear combination of translates of the homogeneous functions.

  • REFINABLE SHIFT INVARIANT SPACES IN ℝd
    International Journal of Wavelets Multiresolution and Information Processing, 2005
    Co-Authors: Carlos Cabrelli, Sigrid B. Heineken, Ursula Molter
    Abstract:

    Let φ : ℝd → ℂ be a compactly supported function which satisfies a refinement equation of the form where Γ ⊂ ℝd is a lattice, Λ is a Finite subset of Γ, and A is a dilation matrix. We prove, under the hypothesis of linear independence of the Γ-translates of φ, that there exists a correspondence between the vectors of the Jordan basis of a Finite submatrix of L = [cAi-j]i,j∈Γ and a Finite-Dimensional Subspace in the shift-invariant space generated by φ. We provide a basis of and show that its elements satisfy a property of homogeneity associated to the eigenvalues of L. If the function φ has accuracy κ, this basis can be chosen to contain a basis for all the multivariate polynomials of degree less than κ. These latter functions are associated to eigenvalues that are powers of the eigenvalues of A-1. Furthermore we show that the dimension of coincides with the local dimension of φ, and hence, every function in the shift-invariant space generated by φ can be written locally as a linear combination of translates of the homogeneous functions.