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.
This paper presents an introduction to geometric representations of quantum states in which each distinct quantum state, pure and mixed, corresponds to a unique point in a Euclidean space. Beginning with a review of some underappreciated properties of the most commonly used geometric representation, the Bloch sphere visualization of qubit states, we show how concepts, algorithms, and spatial relations viewable on this geometric representation can be extended to representations of qudit states of any finite quantum dimension
d and on to the infinite-dimensional limit. A primary goal of the work is helping the reader develop a visual intuition of these spaces, which can complement the understanding of the algebraic formalism of quantum mechanics for learners, teachers, and researchers at any level. Particular emphasis is given both to understanding states in a basis-independent way and to understanding how probability amplitudes and density matrix elements used to algebraically represent states in a particular basis correspond to line segments and angles in the geometric representations. In addition to providing visualizations for such concepts as superpositions, mixtures, decoherence, and measurement, we demonstrate how the representations can be used to substitute simple geometrical calculations for more cumbersome linear algebra ones, which may be of particular use in introducing mixed states and density matrices to beginning quantum students at an early stage. The work concludes with the geometrical interpretation of some commonly used metrics such as the purity of states and their relation to real, Euclidean vectors in the infinite-dimensional limit of the space, which contains all lower-dimensional qudit spaces as subspaces.