scholarly journals The Measure Aspect of Quantum Uncertainty, of Entanglement, and the Associated Entropies

2021 ◽  
Vol 3 (3) ◽  
pp. 534-548
Author(s):  
Ivan Horváth
Keyword(s):  
Number Theory ◽  
Density Matrix ◽  
Quantum State ◽  
Degrees Of Freedom ◽  
Minimal Amount ◽  
Effective Number ◽  
Quantum Uncertainty ◽  
Commuting Operators ◽  
Central Result

Indeterminacy associated with the probing of a quantum state is commonly expressed through spectral distances (metric) featured in the outcomes of repeated experiments. Here, we express it as an effective amount (measure) of distinct outcomes instead. The resulting μ-uncertainties are described by the effective number theory whose central result, the existence of a minimal amount, leads to a well-defined notion of intrinsic irremovable uncertainty. We derive μ-uncertainty formulas for arbitrary set of commuting operators, including the cases with continuous spectra. The associated entropy-like characteristics, the μ-entropies, convey how many degrees of freedom are effectively involved in a given measurement process. In order to construct quantum μ-entropies, we are led to quantum effective numbers designed to count independent, mutually orthogonal states effectively comprising a density matrix. This concept is basis-independent and leads to a measure-based characterization of entanglement.

scholarly journals Effective Number Theory: Counting the Identities of a Quantum State

Entropy ◽  
2020 ◽  
Vol 22 (11) ◽  
pp. 1273
Author(s):  
Ivan Horváth ◽  
Robert Mendris
Keyword(s):  
Number Theory ◽  
Quantum Mechanics ◽  
Quantum Computation ◽  
Quantum State ◽  
Quantum Dynamics ◽  
Quantum Physics ◽  
Quantum Algorithm ◽  
Effective Number ◽  
Additive Theory ◽  
Participation Number

Quantum physics frequently involves a need to count the states, subspaces, measurement outcomes, and other elements of quantum dynamics. However, with quantum mechanics assigning probabilities to such objects, it is often desirable to work with the notion of a “total” that takes into account their varied relevance. For example, such an effective count of position states available to a lattice electron could characterize its localization properties. Similarly, the effective total of outcomes in the measurement step of a quantum computation relates to the efficiency of the quantum algorithm. Despite a broad need for effective counting, a well-founded prescription has not been formulated. Instead, the assignments that do not respect the measure-like nature of the concept, such as versions of the participation number or exponentiated entropies, are used in some areas. Here, we develop the additive theory of effective number functions (ENFs), namely functions assigning consistent totals to collections of objects endowed with probability weights. Our analysis reveals the existence of a minimal total, realized by the unique ENF, which leads to effective counting with absolute meaning. Touching upon the nature of the measure, our results may find applications not only in quantum physics, but also in other quantitative sciences.

Quantum tomograph for measurement and characterization of quantum states of biphoton sources

2020 ◽  
pp. 20-26
Author(s):  
Dmitry N. Frolovtsev ◽  
Sergey A. Magnitskiy ◽  
Andrey V. Demin
Keyword(s):  
Density Matrix ◽  
Quantum State ◽  
Measurement Errors ◽  
Quantum Tomography ◽  
Statistical Characteristics ◽  
Light Sources ◽  
Matrix Elements ◽  
Experimental Implementation

The method and prototype of a device for characterizing of biphoton light sources based on spontaneous parametric downdonversion by quantum tomography are described. The prototype is an experimental implementation of a specialized quantum tomograph designed to measure the quantum polarization states of radiation generated by biphoton sources. Specially developed software will determine the statistical characteristics of the measured quantum state, calculate the tomographic and likelihood estimations of the density matrix, calculate the measurement errors of the density matrix elements and evaluate the quality of the quantum state of biphotons.

Variational characterization of real eigenvalues in linear viscoelastic oscillators

2017 ◽  
Vol 23 (10) ◽  
pp. 1377-1388 ◽  
Author(s):  
Seyyed Abbas Mohammadi ◽  
Heinrich Voss
Keyword(s):  
Degrees Of Freedom ◽  
Nonlinear Eigenvalue Problem ◽  
Eigenvalues And Eigenvectors ◽  
Variational Characterization ◽  
Viscoelastic System ◽  
New Approach ◽  
Motion Equations ◽  
Multiple Degrees Of Freedom ◽  
Linear Viscoelastic

This paper proposes a new approach for computing the real eigenvalues of a multiple-degrees-of-freedom viscoelastic system in which we assume an exponentially decaying damping. The free-motion equations lead to a nonlinear eigenvalue problem. If the system matrices are symmetric, the eigenvalues allow for a variational characterization of maxmin type, and the eigenvalues and eigenvectors can be determined very efficiently by the safeguarded iteration, which converges quadratically and, for extreme eigenvalues, monotonically. Numerical methods demonstrate the performance and the reliability of the approach. The method succeeds where some current approaches, with restrictive physical assumptions, fail.

Unified Rotational and Translational Stiffness Characterization of Compliant Shell Mechanisms

2018 ◽  
Author(s):  
Joost R. Leemans ◽  
Charles J. Kim ◽  
Werner W. P. J. van de Sande ◽  
Just L. Herder
Keyword(s):  
Degrees Of Freedom ◽  
Compliant Mechanisms ◽  
Screw Theory ◽  
Characterization Methods ◽  
Six Degrees Of Freedom ◽  
Thin Walled Structures ◽  
Spatial Mechanisms ◽  
Eigen Decomposition ◽  
Comprehensive Characterization

Compliant shell mechanisms utilize spatially curved thin-walled structures to transfer or transmit force, motion or energy through elastic deformation. To design with spatial mechanisms designers need comprehensive characterization methods, while existing methods fall short of meaningful comparisons between rotational and translational degrees of freedom. This paper presents two approaches, both of which are based on the principle of virtual loads and potential energy, utilizing properties of screw theory, Plücker coordinates and an eigen-decomposition, leading to two unification lengths that can be used to compare and visualize all six degrees of freedom directions and magnitudes of compliant mechanisms in a non-arbitrary physically meaningful manner.

On the Dynamic Isotropy of Mechanisms With Two Degrees of Freedom

2005 ◽  
Author(s):  
Raffaele Di Gregorio ◽  
Alessandro Cammarata ◽  
Rosario Sinatra
Keyword(s):  
Finite Number ◽  
Degrees Of Freedom ◽  
Point Of View ◽  
Closed Curves ◽  
Special Cases ◽  
Dynamic Performances ◽  
Definition Of ◽  
Geometric Locus

The comparison of mechanisms with different topology or with different geometry, but with the same topology, is a necessary operation during the design of a machine sized for a given task. Therefore, tools that evaluate the dynamic performances of a mechanism are welcomed. This paper deals with the dynamic isotropy of 2-dof mechanisms starting from the definition introduced in a previous paper. In particular, starting from the condition that identifies the dynamically isotropic configurations, it shows that, provided some special cases are not considered, 2-dof mechanisms have at most a finite number of isotropic configurations. Moreover, it shows that, provided the dynamically isotropic configurations are excluded, the geometric locus of the configuration space that collects the points associated to configurations with the same dynamic isotropy is constituted by closed curves. This results will allow the classification of 2-dof mechanisms from the dynamic-isotropy point of view, and the definition of some methodologies for the characterization of the dynamic isotropy of these mechanisms. Finally, examples of applications of the obtained results will be given.

scholarly journals Classical Chaos Described by a Density Matrix

Physics ◽  
2021 ◽  
Vol 3 (3) ◽  
pp. 739-746
Author(s):  
Andres Mauricio Kowalski ◽  
Angelo Plastino ◽  
Gaspar Gonzalez
Keyword(s):  
Density Matrix ◽  
Pure State ◽  
Classical Limit ◽  
Degrees Of Freedom ◽  
Temporal Evolution ◽  
Entropy Density ◽  
State Density ◽  
Unexpected Result ◽  
Semiclassical Model ◽  
Classical Chaos

In this paper, a reference to the semiclassical model, in which quantum degrees of freedom interact with classical ones, is considered. The classical limit of a maximum-entropy density matrix that describes the temporal evolution of such a system is analyzed. Here, it is analytically shown that, in the classical limit, it is possible to reproduce classical results. An example is classical chaos. This is done by means a pure-state density matrix, a rather unexpected result. It is shown that this is possible only if the quantum part of the system is in a special class of states.

Design and Fabrication of a Multi-Functional Nanomanipulator

2009 ◽  
Vol 419-420 ◽  
pp. 21-24
Author(s):  
Ming Chang ◽  
Chia Hung Lin ◽  
Chung Po Lin ◽  
Juti Rani Deka
Keyword(s):  
Degrees Of Freedom ◽  
Electrical Characterization ◽  
The Body ◽  
Rapid Expansion ◽  
Nano Materials ◽  
Nano Fabrication ◽  
Permissible Range ◽  
Manipulation System

With rapid expansion of nanotechnology, microminiaturization has become imperative in the field of micro/nano fabrication. A nanomanipulation system with high degrees of freedom that can perform nanomachining, nanofabrication and mechanical/electrical characterization of nanoscale objects inside a scanning electron microscope (SEM) is presented. The manipulation system consists of several individual operating units each having three linear stages and one rotational stage. The body of the manipulator is designed using the idea of superposition. Each operating unit can move in the permissible range of SEM’s vacuum chamber and can increase or decrease the number of units according to the requirement. Experiments were executed to investigate the in-situ electrical resistance of nano materials.

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