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M. Revzen - One of the best experts on this subject based on the ideXlab platform.
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Finite linear Spaces, plane geometries, Hilbert Spaces and finite phase Space
Quantum Studies: Mathematics and Foundations, 2016Co-Authors: M. Revzen, A. MannAbstract:Finite plane geometry is associated with finite Dimensional Hilbert Space. The association allows mapping of q -number Hilbert Space observables to the c -number formalism of quantum mechanics in phase Space. The mapped entities reflect geometrically based line–point interrelation. Particularly simple formulas are involved when use is made of mutually unbiased bases representations for the Hilbert Space entries. The geometry specifies a point–line interrelation. Thus underpinning d -Dimensional Hilbert Space operators (resp. states) with geometrical points leads to operators termed “line operators” underpinned by the geometrical lines. These “line operators”, $$\hat{L}_j;$$ L ^ j ; ( j designates the line) form a complete orthogonal basis for Hilbert Space operators. The representation of Hilbert Space operators in terms of these operators form the phase Space representation of the d -Dimensional Hilbert Space. Examples for the use of the “line operators” in mapping (finite Dimensional) Hilbert Space operators onto finite Dimensional phase Space functions are considered. These include finite Dimensional Wigner function and Radon transform and a geometrical interpretation for the involvement of parity in the mappings of Hilbert Space onto phase Space. Two d -Dimensional particles product states are underpinned with geometrical points. The states, $$|L_j\rangle $$ | L j ⟩ underpinned with the corresponding geometrical lines are maximally entangled states (MES). These “line states” provide a complete $$d^2$$ d 2 Dimensional orthogonal MES basis for for the two d -Dimensional particles. The complete $$d^2$$ d 2 Dimensional MES i.e. the “line states” are shown to provide a transparent geometrical interpretation to the so-called Mean King Problem and its variant. The “line operators” (resp. “line states”) are studied in detail. The paper aims at self sufficiency and to this end all relevant notions are explained herewith.
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a family of weyl wigner transforms for discrete variables defined in a finite Dimensional Hilbert Space
arXiv: Quantum Physics, 2015Co-Authors: A Mann, Pier A Mello, M. RevzenAbstract:We study the Weyl-Wigner transform in the case of discrete variables defined in a Hilbert Space of finite prime-number Dimensionality $N$. We define a family of Weyl-Wigner transforms as function of a phase parameter. We show that it is only for a specific value of the parameter that all the properties we have examined have a parallel with the case of continuous variables defined in an infinite-Dimensional Hilbert Space. A geometrical interpretation is briefly discussed.
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Radon Transform in Finite Dimensional Hilbert Space
EPL (Europhysics Letters), 2012Co-Authors: M. RevzenAbstract:Novel analysis of finite Dimensional Hilbert Space is outlined. The approach bypasses general, inherent, difficulties present in handling angular variables in finite Dimensional problems: The finite Dimensional, d, Hilbert Space operators are underpinned with finite geometry which provide intuitive perspective to the physical operators. The analysis emphasizes a central role for projectors of mutual unbiased bases (MUB) states, extending thereby their use in finite Dimensional quantum mechanics studies. Interrelation among the Hilbert Space operators revealed via their (finite) dual affine plane geometry (DAPG) underpinning are displayed and utilized in formulating the finite Dimensional ubiquitous Radon transformation and its inverse illustrating phase Space-like physics encoded in lines and points of the geometry. The finite geometry required for our study is outlined.
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Geometrical Underpinning of Finite Dimensional Hilbert Space
arXiv: Quantum Physics, 2011Co-Authors: M. RevzenAbstract:Finite geometry is employed to underpin operators in finite, d, Dimensional Hilbert Space. The central role of mutual unbiased bases (MUB) states projectors is exhibited. Interrelation among operators in Hilbert Space, revealed through their (finite) dual affine plane geometry (DAPG) underpinning is studied. Transcription to (finite) affine plane geometry (APG) is given and utilized for their interpretation.
Alain Haraux - One of the best experts on this subject based on the ideXlab platform.
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Existence, uniqueness and global behavior of the solutions to some nonlinear vector equations in a finite Dimensional Hilbert Space
Nonlinear Analysis-theory Methods & Applications, 2017Co-Authors: Mama Abdelli, María Anguiano, Alain HarauxAbstract:Abstract The initial value problem and global properties of solutions are studied for the vector equation: ( ‖ u ′ ‖ l u ′ ) ′ + ‖ A 1 2 u ‖ β A u + g ( u ′ ) = 0 in a finite Dimensional Hilbert Space under suitable assumptions on g .
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Existence, uniqueness and global behavior of the solutions to some nonlinear vector equations in a finite Dimensional Hilbert Space
2016Co-Authors: Mama Abdelli, María Anguiano, Alain HarauxAbstract:The initial value problem and global properties of solutions are studied for the vector equation: $\Big(\|u'\|^{l}u'\Big)'+\|A^{\frac{1}{2}}u\|^\beta Au+g(u')=0$ in a finite Dimensional Hilbert Space under suitable assumptions on $g$.
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Existence, uniqueness and global behavior of the solutions to some nonlinear vector equations in a finite Dimensional Hilbert Space
arXiv: Classical Analysis and ODEs, 2015Co-Authors: Mama Abdelli, María Anguiano, Alain HarauxAbstract:The initial value problem and global properties of solutions are studied for the vector equation: $\left(\left\Vert u'\right\Vert ^ l u'\right)' + \left\Vert A^{\frac{1}{2}} u \right\Vert^\beta A u + g(u ') = 0$ in a finite Dimensional Hilbert Space under suitable assumptions on g.
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The Łojasiewicz gradient inequality in the infinite-Dimensional Hilbert Space framework
Journal of Functional Analysis, 2011Co-Authors: Alain Haraux, Mohamed Ali JendoubiAbstract:Abstract We provide a reasonably optimal answer to the natural question of the conditions under which an analytic function on an infinite-Dimensional Hilbert Space satisfies the Łojasiewicz gradient inequality.
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The Lojasiewicz gradient inequality in the infinite Dimensional Hilbert Space framework
2010Co-Authors: Alain Haraux, Mohamed Ali JendoubiAbstract:Abstract: We provide a reasonably optimal answer to the natural question of the conditions under which an analytic function on an infinite Dimensional Hilbert Space satisfies the Lojasiewicz gradient inequality.
Adam Miranowicz - One of the best experts on this subject based on the ideXlab platform.
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quantum optical states in finite Dimensional Hilbert Space ii state generation
arXiv: Quantum Physics, 2002Co-Authors: W Leonski, Adam MiranowiczAbstract:In the second part of our review (for the first part see quant-ph/0108080), we discuss a physical model for generation of "truncated" coherent and squeezed states in finite-Dimensional Hibert Spaces.
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quantum optical states in finite Dimensional Hilbert Space i general formalism
arXiv: Quantum Physics, 2002Co-Authors: Adam Miranowicz, W Leonski, Nobuyuki ImotoAbstract:The interest in quantum-optical states confined in finite-Dimensional Hilbert Spaces has recently been stimulated by the progress in quantum computing, quantum-optical state preparation, and measurement techniques, in particular, by the development of the discrete quantum-state tomography. In the first part of our review we present two essentially different approaches to define harmonic oscillator states in the finite-Dimensional Hilbert Spaces. One of them is related to the truncation scheme of Pegg, Phillips and Barnett [Phys. Rev. Lett. 81, 1604 (1998)] -- the so-called quantum scissors device. The second method corresponds to the truncation scheme of Leo\'nski and Tana\'s [Phys. Rev. A 49, R20 (1994)]. We propose some new definitions of the states related to these truncation schemes and find their explicit forms in the Fock representation. We discuss finite-Dimensional generalizations of coherent states, phase coherent states, displaced number states, Schr\"odinger cats, and squeezed vacuum. We show some intriguing properties of the states with the help of the discrete Wigner function.
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Harmonic Oscillator States in Finite Dimensional Hilbert Space
Quantum Optics and the Spectroscopy of Solids, 1997Co-Authors: Adam Miranowicz, Tomáš Opatrný, Jiří BajerAbstract:We consider various harmonic-oscillator states in a finite-Dimensional Hilbert Space. In greater detail we analyze finite-Dimensional coherent states defined either by truncation of a number-state expansion of the standard coherent states or by the action of the generalized finite-Dimensional displacement operator on vacuum. Within these two approaches other finite-Dimensional states are analyzed, including displaced number states, and even and odd coherent states. We propose some new definitions of the states in a finiteDimensional Hilbert Space and construct explicitly all these states in Fock representation. The number-phase Wigner function is computed for several states. Analytical results and numerically computed graphs are presented.
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quantum state engineering in finite Dimensional Hilbert Space
Acta Physica Slovaca, 1996Co-Authors: Adam Miranowicz, W Leonski, S Dyrting, R TanaśAbstract:We give a recipe for how to generate various harmonic oscillator states formally defined in finite-Dimensional Hilbert Space.
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Coherent states in finite-Dimensional Hilbert Space and their Wigner representation
Journal of Modern Optics, 1996Co-Authors: Tomáš Opatrný, Adam Miranowicz, Jiří BajerAbstract:We consider two definitions of coherent states in a finite-Dimensional Hilbert Space based on (i) truncation of the usual coherent state expansion and (ii) generalization of the displacement operator acting on vacuum. The number-phase Wigner function is computed for such states. Analytical results and numerically computed graphs are presented. Special attention is paid to two-level states and to their Stokes parameter representations.
Yeong-cheng Liou - One of the best experts on this subject based on the ideXlab platform.
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strong convergence of a modified extragradient method to the minimum norm solution of variational inequalities
Abstract and Applied Analysis, 2012Co-Authors: Muhammad Aslam Noor, Yeong-cheng LiouAbstract:We suggest and analyze a modified extragradient method for solving variational inequalities, which is convergent strongly to the minimum-norm solution of some variational inequality in an infinite-Dimensional Hilbert Space.
Mang Feng - One of the best experts on this subject based on the ideXlab platform.
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quantum algorithms and mathematical formulations of biomolecular solutions of the vertex cover problem in the finite Dimensional Hilbert Space
IEEE Transactions on Nanobioscience, 2015Co-Authors: Wenglong Chang, Tingting Ren, Mang FengAbstract:In this paper, it is shown that the proposed quantum algorithm for implementing Boolean circuits generated from the DNA-based algorithm solving the vertex-cover problem of any graph ${\mbi G}$ with ${\mbi m}$ edges and ${\mbi n}$ vertices is the optimal quantum algorithm. Next, it is also demonstrated that mathematical solutions of the same biomolecular solutions are represented in terms of a unit vector in the finite-Dimensional Hilbert Space. Furthermore, for testing our theory, a nuclear magnetic resonance (NMR) experiment of three quantum bits to solve the simplest vertex-cover problem is completed.