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.
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.
Finite-dimensional density matrices contain structural information that is not exhausted by their spectra. This work develops a framework in two parts. The first establishes the intrinsic reference basis (IRB), a real orthonormal frame canonically determined by the real symmetric part of the density matrix, whose basis states are called principal population-coherence states, or intrinsic resolving states. In this frame the population content is diagonal and ordered, while the total imaginary cohesion, measured by the Frobenius norm of the antisymmetric block, is preserved. The IRB therefore separates populations from coherences without converting coherence into basisdependent phases, a property the eigenbasis cannot supply because it sends imaginary coherences to zero by construction. The second part defines geometric nonregularity as the excess cohesion that remains after the dominant pure component of the discriminating decomposition has been subtracted. Regularity is recovered as the limiting case in which the discriminant component carries no imaginary cohesion. The normalization of the degree of geometric nonregularity is fixed by the maximally nonregular discriminants, which exhibit a sharp dimensional transition: in dimension three (qutrits and three dimensional polarization states) the maximizer features a circular-pluslinear structure, whereas in dimensions four and higher the maximizer admits a doubly-coherent two-sector structure. In three dimensions the excess cohesion is moreover dual to an excess spin vector, providing directional information that scalar measures discard. The framework reconciles the original scalar definition of nonregularity with a geometric, representation-intrinsic formulation and exposes physical content of finite-dimensional density matrices that the eigenvalues alone do not reveal.
The intrinsic representation of a finite-dimensional density matrix splits it, in the real ortogonal frame that diagonalizes its real part, into a diagonal block of intrinsic populations and an antisymmetric block of intrinsic coherences. The populations admit a complete hierarchical description through the indices of population asymmetry, whereas the coherence sector has so far had only a single global descriptor. We construct, in arbitrary dimension, a factorization of the intrinsic density matrix into a diagonal population factor and a population-isotropic core whose imaginary part is a real antisymmetric object, the metaspin tensor. Its skew-singular values are bounded above by unity, generalizing a bound known in three dimensions for polarization states and qutrits, and from them a nested family of indices of correlation asymmetry is built, mirroring the indices of population asymmetry. The construction reproduces the metaspin vector in three dimensions and, in higher dimensions, yields a finer classification of nonregular states, reinterpreting the known dimensional transition of nonregularity as a change in the saturation pattern of the metaspin tensor. The framework provides a real-orthogonal intrinsic hierarchical descriptor of the imaginary part of a density matrix, with applications to optical polarimetry, electromagnetic polarization matrices, and the structural analysis of quantum states in arbitrary dimension.
Abstract By using the recently introduced parametrization of an n-dimensional density matrix in terms of the indices of population asymmetry and the intrinsic coherences, we define descriptors in both integer and continuous forms of the effective dimension that take place for a complete description of a density matrix, thus providing accurate information beyond the rank of the density matrix. The concepts of dimensional folding, hidden dimensional purity, and dimensional entropy are introduced and discussed in view of the new approach presented. The results are applicable to any physical system represented by a density matrix, such as n-level quantum systems, qutrits, sets of interacting pencils of radiation, classical polarization states, and to transformations of density matrices, as occurs with quantum channels.
Finite-dimensional density matrices contain two representation-intrinsic sectors after the real part is diagonalized, namely ordered intrinsic populations and antisymmetric imaginary coherences. This article develops exact complementarity identities showing how these sectors determine purity, spectral concentration, and entropy. Populations are described by indices of population asymmetry, while coherences are described by the Youla spectrum of the dimensionless metaspin tensor and by correlation-asymmetry indices. In the aligned class, where Youla two-planes coincide with pairs of intrinsic axes, normalized purity splits into a population hierarchy and pairwise coherence terms weighted by products of intrinsic populations. For arbitrary orientations, the coherence term is expressed as a positive semi-definite bilinear form in population-weighted Plücker coordinates. For fixed populations and pairing, increasing any Youla value sharpens the spectrum by majorization and decreases all Rényi entropies, including the von Neumann limit. For fixed ordered populations, maximum aligned cohesion is obtained by saturating adjacent population pairs. The dimensional transition of the discriminating-component cohesion bound is then interpreted as the change from one to two simultaneously saturating metaspin pairs.
Several approaches and descriptors have been proposed to characterize the purity of coherency or density matrices describing physical states, including the polarimetric purity of 2D and 3D partially polarized waves. This work introduces two interpretations of the degree of purity: one derived from statistics and another from algebra. In the first one, the degree purity is expressed in terms of the mean and standard deviation of the eigenvalue spectrum of the density or coherency matrix of the corresponding state. The second one expresses the purity in terms of two specific measures obtained by decomposing the coherency matrix as a sum of traceless symmetric, antisymmetric, and scalar matrices. We believe these two approaches offer better insights into the purity measure. Furthermore, interesting relations with existing quantities in polarization optics also are described.
There are many coincidences that relate the treatment of quantum states in a two-dimensional space and the classical polarisation treatment, due to some similarities that occur in coherence matrix and density matrix.We use these similarities to separate the density matrix in a mixture of a pure state and a maximally mixed state.This mixture is controlled by the degree of purity, that measures how much pure is a quantum state, such as usually done with the coherence matrix of a partially polarised beam.After that, we use this degree of purity in quantum information theory.We find out its relationship with the von Neumann entropy and its use in quantum Kolmogorov distance, fidelity and mutual information.