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.

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 Insights into one-body density matrices using deep learning

2020 ◽  
Vol 224 ◽  
pp. 265-291 ◽  
Author(s):  
Jack Wetherell ◽  
Andrea Costamagna ◽  
Matteo Gatti ◽  
Lucia Reining
Keyword(s):  
Deep Learning ◽  
Density Matrix ◽  
Reduced Density Matrix ◽  
Body Density ◽  
Density Matrices ◽  
Learning Constraints ◽  
The One ◽  
Reduced Density

Deep-learning constraints of the one-body reduced density matrix from its compressibility to enable efficient determination of key observables.

Is it Reasonable to Obtain Information on the Polarizability and Hyperpolarizability Only from the Electron Density?

2018 ◽  
Vol 71 (4) ◽  
pp. 295 ◽  
Author(s):  
Dylan Jayatilaka ◽  
Kunal K. Jha ◽  
Parthapratim Munshi
Keyword(s):  
Density Matrix ◽  
Electron Density ◽  
Ground State ◽  
Density Functional ◽  
Particle Density ◽  
Density Matrices ◽  
Hartree Fock ◽  
Electron Density Matrix ◽  
Ground State Electron ◽  
Side Result

Formulae for the static electronic polarizability and hyperpolarizability are derived in terms of moments of the ground-state electron density matrix by applying the Unsöld approximation and a generalization of the Fermi-Amaldi approximation. The latter formula for the hyperpolarizability appears to be new. The formulae manifestly transform correctly under rotations, and they are observed to be essentially cumulant expressions. Consequently, they are additive over different regions. The properties of the formula are discussed in relation to others that have been proposed in order to clarify inconsistencies. The formulae are then tested against coupled-perturbed Hartree-Fock results for a set of 40 donor-π-acceptor systems. For the polarizability, the correlation is reasonable; therefore, electron density matrix moments from theory or experiment may be used to predict polarizabilities. By constrast, the results for the hyperpolarizabilities are poor, not even within one or two orders of magnitude. The formula for the two- and three-particle density matrices obtained as a side result in this work may be interesting for density functional theories.

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.

Some Features of Excited States Density Matrix Calculation and Their Pairing Relations in Conjugated Systems

1983 ◽  
Vol 38 (5) ◽  
pp. 595-600
Author(s):  
Myriam Segre de Giambiagi ◽  
Mario Giambiagi
Keyword(s):  
Density Matrix ◽  
Excited States ◽  
Simple Approach ◽  
Alternative Formulation ◽  
Density Matrices ◽  
Convergence Problem ◽  
Conjugated Systems ◽  
Self Consistent ◽  
Natural Separation ◽  
Matrix Calculation

Direct PPP-type calculations of self-consistent (SC) density matrices for excited states are described and the corresponding “thawn” molecular orbitals (MO) are discussed. Special atten­tion is addressed to particular solutions arising in conjugated systems of a certain symmetry, and to their chemical implications. The U(2) and U(3) algebras are applied, respectively, to the 4- electron and 6-electron cases; a natural separation of excited states in different cases follows. A simple approach to the convergence problem for excited states is given. The complementarity relations, an alternative formulation of the pairing theorem valid for heteromolecules and non-alternant systems, allow some fruitful experimental applications. Together with the extended pairing relations shown here, they may help to rationalize general trends.

CANONICAL FORM AND SEPARABILITY OF PPT STATES IN ${\mathcal C}^2\otimes {\mathcal C}^M\otimes {\mathcal C}^N$ COMPOSITE QUANTUM SYSTEMS

2003 ◽  
Vol 01 (03) ◽  
pp. 337-347
Author(s):  
XIAO-HONG WANG ◽  
SHAO-MING FEI ◽  
ZHI-XI WANG ◽  
KE WU
Keyword(s):  
Canonical Form ◽  
General Expression ◽  
Quantum Systems ◽  
Canonical Forms ◽  
Density Matrices ◽  
Partial Transposition ◽  
Separability Condition ◽  
Ppt States

We investigate the canonical forms of positive partial transposition (PPT) density matrices in [Formula: see text] composite quantum systems with rank N. A general expression for these PPT states are explicitly obtained. From this canonical form a sufficient separability condition is presented.

scholarly journals Distinguishability of non-orthogonal density matrices does not imply violations of the second law

2019 ◽  
Author(s):  
PierGianLuca Porta Mana
Keyword(s):  
Hilbert Space ◽  
Density Matrix ◽  
Thermodynamic Analysis ◽  
Second Law Of Thermodynamics ◽  
The Other ◽  
Second Law ◽  
Density Matrices ◽  
Matrix Space ◽  
Von Neumann ◽  
Quantum Thermodynamic

The hypothetical possibility of distinguishing preparations described by non-orthogonal density matrices does not necessarily imply a violation of the second law of thermodynamics, as was instead stated by von Neumann. On the other hand, such a possibility would surely mean that the particular density-matrix space (and related Hilbert space) adopted would not be adequate to describe the hypothetical new experimental facts. These points are shown by making clear the distinction between physical preparations and the density matrices which represent them, and then comparing a "quantum" thermodynamic analysis given by Peres with a "classical" one given by Jaynes.

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