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

Abstract

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.

Extending the degree of polarization concept to higher-order and orbital angular momentum Poincaré spheres

“…In addition, parameter p in equation (17), which controls the weight of each mode in the incoherent superposition, is obtained as the length of the OAM corresponding Stokes vector and its value is expected to be same as the DoP of the input beam to the qplate-polarizer system. Note that this value can be regarded as a degree of purity in the equivalent Bloch sphere [36,37].…”

“…This theoretical framework is presented in section 2. We show that the input DoP is maintained on the vector beam after the q-plate and that it can be regarded as an equivalent degree of purity of the superposition of pure modes on the OAMPS, here considered as the length of the Bloch vector of the simplest two-dimensional Bloch sphere [36,37]. The examples are illustrated using the well-known Laguerre-Gaussian beams, although this formalism can be used to describe other beams with OAM.…”

Geometric descriptions for the polarization of nonparaxial light: a tutorial

“…Generalizations to the different measures of polarization to higher dimensions have been discussed by several authors [81,91,121,122]. The barycentric interpretation for the measures of degree of polarization is also easily generalized by using N point masses whose magnitudes are the eigenvalues Λ i , and which are located at a unit distance from the origin and each equidistant to all other masses.…”

Geometric Interpretation and General Classification of Three-Dimensional Polarization States through the Intrinsic Stokes Parameters

“…In this work, the geometric representation introduced by Dennis [33] for general three-dimensional (3D) polarization states is studied and interpreted in terms of the intrinsic Stokes parameters [21] and other meaningful descriptors that are invariant under rotations of the reference frame [17][18][19][20][21][22][33][34][35][36][37][38][39][40][41][42][43][44][45]. The classification introduced in Refs.…”

“…The classification introduced in Refs. [20,46] is improved and completed in the light of the recent approaches on nonregularity [39,42,45,47,48], polarimetric dimension [41], spin of a polarization state [43,47,49] and interpretation of sets of orthogonal 3D polarization states [44].…”