scholarly journals Sources of Asymmetry and the Concept of Nonregularity of n-Dimensional Density Matrices

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1002
Author(s):  
José J. Gil
Keyword(s):  
Density Matrix ◽  
Reference Frame ◽  
Physical Nature ◽  
Pure Component ◽  
Stokes Parameters ◽  
Density Matrices ◽  
Polarization Density ◽  
Intimate Link ◽  
Nd Systems ◽  
Definition Of

The information contained in an n-dimensional (nD) density matrix ρ is parametrized and interpreted in terms of its asymmetry properties through the introduction of a family of components of purity that are invariant with respect to arbitrary rotations of the nD Cartesian reference frame and that are composed of two categories of meaningful parameters of different physical nature: the indices of population asymmetry and the intrinsic coherences. It is found that the components of purity coincide, up to respective simple coefficients, with the intrinsic Stokes parameters, which are also introduced in this work, and that determine two complementary sources of purity, namely the population asymmetry and the correlation asymmetry, whose weighted square average equals the overall degree of purity of ρ. A discriminating decomposition of ρ as a convex sum of three density matrices, viz. the pure, the fully random (maximally mixed) and the discriminating component, is introduced, which allows for the definition of the degree of nonregularity of ρ as the distance from ρ to a density matrix of a system composed of a pure component and a set of 2D, 3D,… and nD maximally mixed components. The chiral properties of a state ρ are analyzed and characterized from its intimate link to the degree of correlation asymmetry. The results presented constitute a generalization to nD systems of those established and exploited for polarization density matrices in a series of previous works.

Über die Symmetrie-Eigenschaften der reduzierten Dichtematrizen und der natürlichen Spin-Orbitale und Spin-Geminale (der natürlichen Ein- und Zwei-Elektronen-Funktionen)

1963 ◽  
Vol 18 (10) ◽  
pp. 1058-1064 ◽  
Author(s):  
Werner Kutzelnigg
Keyword(s):  
Wave Function ◽  
Density Matrix ◽  
Symmetry Group ◽  
Total Wave Function ◽  
Irreducible Representations ◽  
Density Matrices ◽  
Reduced Density Matrices ◽  
Total Wave ◽  
Definition Of ◽  
Reduced Density

The density operator (density matrix) of a quantum mechanical system can be decomposed into operators which transform as irreducible representations of the symmetry group in coordinate and spin space. Each of these components has a physical meaning connected with the expectation values of certain operators. The reduced density matrices can be decomposed in a completely analogous way.The symmetry properties of the total wave function give rise to degeneracies of the eigenvalues of the reduced density matrices. These degeneracies can be removed by requiring that the natural spin orbitals (NSO, defined as the eigenfunctions of the first order density matrix), as well as the natural spin geminais (NSG, the eigenfunctions of the second order density matrix) and their spinless counterparts transform as irreducible representations of the symmetry group and are eigenfunctions of S2 and Sz.In many important cases this requirement is compatible with the original definition of the NSO, the NSG etc. e. g., when there is no spatial degeneracy of the total wave function and when the Z-component of the total spin vanishes. When these conditions are not fulfilled an alternative definition of the NSO and the NSG is proposed.

Modern Conception of Combined “ICS”-approach (“Information, Cybernetic and Synergetic”) for Analysis and Optimization of Legal Ergasystems

2020 ◽  
Vol 2 (1) ◽  
pp. 59-81
Author(s):  
D. A. Lovtsov ◽  
Keyword(s):  
Public Relations ◽  
Functional Organization ◽  
Dynamic Equilibrium ◽  
Physical Nature ◽  
Legal Regulation ◽  
Cybernetic System ◽  
Methodological Principles ◽  
Definition Of ◽  
Concept Of Information ◽  
Theoretical Apparatus

Introduction. The lack of a coherent systemology law does not enable the use of evidence-based formalization to solve the basic theoretical problems of law interpretation and enforcement. The development of an appropriate formal-theoretical apparatus is possible on the basis of a productive systemological concept. The justification of this concept is based on the study of philosophical bases and fundamental principles (integrity, dynamic equilibrium, feedback, etc.) and the use of logical and linguistic methods of problem-oriented system approach. Theoretical Basis. Methods. The conceptual and logical modeling of legal ergasystems, the systems analysis and resolution of the theory-applied base of technology of two-tier legal regulation; the synthesis and modification of private scientific results of the author published in 2000–2019, with copyright in the author’s scientific works and educational publications. Results. The contemporary conceptual variant of combined “ICS”-approach (“information, cybernetic and synergetic”) as a general methodology of analysis and optimization of legal ergasystems, as characterized by the following conditions: the substantiation of the appropriate three-part set of methodological research principles, corresponding to the triple-aspect physical nature of the study of complex legal systems as ergasystems; the clarification of the conceptual and logical model of the legal ergasystem taking into account the fundamental feedback principle; the definition of the law of necessary diversity of William R. Ashby is justified and corresponding conditions of realize of effective technology of two-level (normative and individual) legal regulation; the definition of basic concepts and methodological principles of modern systemology of legal regulation; the justification of the functional organization of the Invariant Rational Control Loop. Discussion and Conclusion. A developed conceptual object-oriented version of combined “ICS”-approach for analysis and optimization of legal ergasystems is a methodological basis for the development of a working formal-theoretical apparatus of legal regulation systemology. This will formalize the decisions of the main theoretical problems of law interpretation and enforcement, as well as developing and implementing special information and legal technologies based on the concept of information and functional databases and knowledge. This will in turn ensure the information increases the effectiveness of the system of legal regulation of public relations as an information and cybernetic system subject to the subjective organizing process of human activity and the objective synergetic processes of disorganization.

scholarly journals The Celestial Reference System in Relativistic Framework

1990 ◽  
Vol 141 ◽  
pp. 99-110
Author(s):  
Han Chun-Hao ◽  
Huang Tian-Yi ◽  
Xu Bang-Xin
Keyword(s):  
Coordinate System ◽  
Reference Frame ◽  
Reference System ◽  
Light Deflection ◽  
Coordinate Systems ◽  
Definition Of ◽  
Celestial Sphere ◽  
Celestial Reference System ◽  
Relativistic Framework

The concept of reference system, reference frame, coordinate system and celestial sphere in a relativistic framework are given. The problems on the choice of celestial coordinate systems and the definition of the light deflection are discussed. Our suggestions are listed in Sec. 5.

N-Representable one-electron reduced density matrices reconstruction at non-zero temperatures

2021 ◽  
Vol 77 (5) ◽  
Author(s):  
Yoann Launay ◽  
Jean-Michel Gillet
Keyword(s):  
Density Matrix ◽  
Thermal Effects ◽  
Scattering Data ◽  
Temperature Dependent ◽  
Density Matrices ◽  
Reduced Density Matrices ◽  
Structure Factors ◽  
Reduced Density ◽  
Statistical Noise

This article retraces different methods that have been explored to account for the atomic thermal motion in the reconstruction of one-electron reduced density matrices from experimental X-ray structure factors (XSF) and directional Compton profiles (DCP). Attention has been paid to propose the simplest possible model, which obeys the necessary N-representability conditions, while accurately reproducing all available experimental data. The deconvolution of thermal effects makes it possible to obtain an experimental static density matrix, which can directly be compared with theoretical 1-RDM (reduced density matrix). It is found that above a 1% statistical noise level, the role played by Compton scattering data becomes negligible and no accurate 1-RDM is reachable. Since no thermal 1-RDM is available as a reference, the quality of an experimentally derived temperature-dependent matrix is difficult to assess. However, the accuracy of the obtained static 1-RDM, through the performance of the refined observables, is strong evidence that the Semi-Definite Programming method is robust and well adapted to the reconstruction of an experimental dynamical 1-RDM.

scholarly journals The permanent tide and the International Height Reference Frame IHRF

2021 ◽  
Vol 95 (9) ◽  
Author(s):  
Jaakko Mäkinen
Keyword(s):  
Reference Frame ◽  
International Association ◽  
Reference Value ◽  
Equipotential Surface ◽  
Terrestrial Reference Frame ◽  
Reference Level ◽  
Consistent Estimate ◽  
The Mean ◽  
Global Mean Sea Level ◽  
Definition Of

AbstractThe International Height Reference System (IHRS), adopted by International Association of Geodesy (IAG) in its Resolution No. 1 at the XXVI General Assembly of the International Union of Geodesy and Geophysics (IUGG) in Prague in 2015, contains two novelties. Firstly, the mean-tide concept is adopted for handling the permanent tide. While many national height systems continue to apply the mean-tide concept, this was the first time that the IAG officially introduced it for a potential field quantity. Secondly, the reference level of the height system is defined by the equipotential surface where the geopotential has a conventional value W0 = 62,636,853.4 m2 s–2. This value was first determined empirically to provide a good approximation to the global mean sea level and then adopted as a reference value by convention. I analyse the tidal aspects of the reference level based on W0. By definition, W0 is independent of the tidal concept that was adopted for the equipotential surface, but for different concepts, different functions are involved in the W of the equation W = W0. I find that, in the empirical determination of the adopted estimate W0, the permanent tide is treated inconsistently. However, the consistent estimate from the same data rounds off to the same value. I discuss the tidal conventions and formulas for the International Height Reference Frame (IHRF) and the realisation of the IHRS. I propose a simplified definition of IHRF geopotential numbers that would make it possible to transform between the IHRF and zero-tide geopotential numbers using a simple datum-difference surface. Such a transformation would not be adequate if rigorous mean-tide formulas were imposed. The IHRF should adopt a conventional (best) estimate of the permanent tide-generating potential, such as that which is contained in the International Earth Rotation and Reference Systems Service Conventions, and use it as a basis for other conventional formulas. The tide-free coordinates of the International Terrestrial Reference Frame and tide-free Global Geopotential Models are central in the modelling of geopotential for the purposes of the IHRF. I present a set of correction formulas that can be used to move to the zero-tide model before, during, or after the processing, and finally to the mean-tide IHRF. To reduce the confusion around the multitude of tidal concepts, I propose that modelling should primarily be done using the zero-tide concept, with the mean-tide potential as an add-on. The widespread use of the expression “systems of permanent tide” may also have contributed to the confusion, as such “systems” do not have the properties that are generally associated with other “systems” in geodesy. Hence, this paper mostly uses “concept” instead of “system” when referring to the permanent tide.

Does symmetry Imply PPT Property?

2015 ◽  
Vol 15 (9&10) ◽  
pp. 812-824
Author(s):  
Daniel Cariello
Keyword(s):  
Density Matrix ◽  
Tensor Rank ◽  
Natural Question ◽  
Density Matrices ◽  
Partial Transposition ◽  
Class Of Matrices ◽  
Positive Coefficients

Recently it was proved that many results that are true for density matrices which are positive under partial transposition (or simply PPT), also hold for another class of matrices with a certain symmetry in their Hermitian Schmidt decompositions. These matrices were called symmetric with positive coefficients (or simply SPC). A natural question appeared: What is the connection between SPC matrices and PPT matrices? Is every SPC matrix PPT? Here we show that every SPC matrix is PPT in $M_2\otimes M_2$. This theorem is a consequence of the fact that every density matrix in $M_2\otimes M_m$, with tensor rank smaller or equal to 3, is separable. Although, in $M_3\otimes M_3$, we present an example of SPC matrix with tensor rank 3 that is not PPT. We shall also provide a non trivial example of a family of matrices in $M_k\otimes M_k$, in which both, the SPC and PPT properties, are equivalent. Within this family, there exists a non trivial subfamily in which the SPC property is equivalent to separability.

scholarly journals Contributions of the USNO to the Optical Reference Frame

1997 ◽  
Vol 165 ◽  
pp. 453-462
Author(s):  
Thomas Corbin
Keyword(s):  
Reference Frame ◽  
Reverse Order ◽  
Good Working ◽  
Working Definition ◽  
Optical Reference ◽  
Definition Of ◽  
Celestial Sphere ◽  
Celestial Reference Frame ◽  
Frame Rigidity ◽  
Zero Points

A good, working definition of what is required in a celestial reference frame is that it must provide observable fiducial points on the Celestial Sphere with internally consistent positions that are referred to coordinate axes of known direction. In reality, this statement gives the goals in the reverse order from that in which each must be achieved, the definition of the axes, or zero points of the system give orientation to the observationally defined set of primary objects whose coordinate relation to each other must give the frame rigidity. Finally, the primary objects are generally too sparse to define the frame within areas of less than tens of square degrees, and so additional objects must be related to the frame to increase the density. This last step is required to make the frame useful for most observational applications.

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