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.
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.
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.
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.
We present a geometric approach to characterize quantum coherence through entropic diagrams that map the eigenvalue spectra of density matrices into a two-dimensional space defined by von Neumann and Tsallis-2 entropies. We derive the relative entropy of coherence using the Schur-Horn majorization theorem and introduce a new coherence monotone—the relative cross-entropy of coherence—which is valid only for mixed states and whose behavior under various classes of quantum operations is examined in detail. Interestingly, our approach reveals distinct entropy-coherence distillation curves and introduces the notion of degenerate coherence distillation, where coherence is concentrated into higher-dimensional, partially uniform spectral configurations. Although the degenerate coherence distillation does not always surpass standard qubit coherence distillation in asymptotic rate, it may offer an operationally meaningful alternative for protocols that require high-dimensional or symmetrically structured coherence resources. In addition, we derive a refined entropic uncertainty relation that incorporates the maximum eigenvalue of post-measurement states, which leads to state-dependent bounds that improve upon the conventional Maassen-Uffink bound. These findings suggest that entropic diagrams may serve not only as a useful visual and analytical tool but also as a guide for developing resource-efficient strategies in quantum information processing.
Various measures have been suggested recently for quantifying the coherence of a quantum state with respect to a given basis. We first use two of these, the l1-norm and relative entropy measures, to investigate tradeoffs between the coherences of mutually unbiased bases. Results include relations between coherence, uncertainty, and purity; tight general bounds restricting the coherences of mutually unbiased bases; and an exact complementarity relation for qubit coherences. We further define the average coherence of a quantum state. For the l1-norm measure this is related to a natural ``coherence radius'' for the state and leads to a conjecture for an l2-norm measure of coherence. For relative entropy the average coherence is determined by the difference between the von Neumann entropy and the quantum subentropy of the state and leads to upper bounds for the latter quantity. Finally, we point out that the relative entropy of coherence is a special case of G-asymmetry, which immediately yields several operational interpretations in contexts as diverse as frame alignment, quantum communication, and metrology, and suggests generalizing the property of quantum coherence to arbitrary groups of physical transformations.
-dimensional density matrices, we characterize each state by a normalized global purity index and two complementary indices, which quantify the contributions of level populations and coherences. This yields a simple quadratic relation and a geometric representation in a "population-coherence plane", where different routes to purity can be distinguished. In the two-level case, we construct explicit families of states with identical spectra and global purity but opposite internal structure, realizing population-dominated and coherence-dominated routes. We then apply this framework to a standard Page-type evaporation model without an explicit Hamiltonian, in which a black hole and its Hawking radiation form a bipartite pure state with varying Hilbert-space dimensions. Using known results for typical reduced states in large dimensions, we analyze the behavior of population and coherence components of purity along the evaporation process. Under the physically motivated requirement that, in this energy-free setting, the radiation populations remain nearly uniform in the chosen basis, we show that the late-time recovery of purity must be coherence-dominated: the global purity of the radiation approaches unity while the population index stays small and the coherence index carries essentially all the purity.
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.