Spectral-angular parametrization of open qudit dynamics

Quantum Physics [Submitted on 13 Apr 2026] Spectral-angular parametrization of open qudit dynamics Jean-Pierre Gazeau, Kaoutar El Bachiri, Zakaria Bouameur, Yassine Hassouni We present a parametrization of density matrices (mixed states) in a finite-dimensional Hilbert space \mathbb{C}^n, particularly suited to the description of their time evolution as open quantum systems governed by GKLS dynamics. A generic (non-degenerate) density matrix rho_{\mathbf{r},\pmb{\phi}}, characterized by n^2-1 real parameters, naturally decomposes into two sets: (i) an (n-1)-tuple \mathbf{r} of spectral parameters, constrained to lie in a convex polytope, and (ii) a set of n^2-n angular variables \pmb{\phi}, associated with the flag manifold \simeq \mathrm{SU}(n)/\mathbb{T}^{n-1}, where \mathbb{T}^{n-1} is the standard maximal diagonal torus, in the spirit of the Tilma--Sudarshan construction. A key observation is that the spectral parameters \mathbf{r} = (r_1, \ldots, r_{n-1}) admit a natural Lie-algebraic interpretation: they are precisely the simple root coordinates of the eigenvalue vector in the Cartan subalgebra of A_{n-1} = \mathfrak{sl}(n), with each r_i = p_i - p_{i+1} corresponding to the simple root \alpha_i = e_i - e_{i+1}. The convex polytope constraining \mathbf{r} is thus the positive Weyl chamber of A_{n-1}, and the full spectral domain R_{n-1} is the corresponding weight polytope. This parametrization leads to a partial decoupling of the dynamics: the evolution of the angular variables depends on both the Hamiltonian and the dissipative part of the Lindblad generator, whereas the evolution of the spectral parameters involves only the dissipative contribution. Low-dimensional examples for n=2 and n=3 are discussed in detail, including an application to the trichromatic structure of human colour perception, and we propose an alternative definition of purity expressed solely in terms of the spectral parameters \mathbf{r}. Comments: 30 pages, 4 figures Subjects: Quantum Physics (quant-ph) MSC classes: 81P16, 81R05, 81R30, 81S22 Cite as: arXiv:2604.11864 [quant-ph]   (or arXiv:2604.11864v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2604.11864 Focus to learn more Submission history From: Jean Pierre Gazeau [view email] [v1] Mon, 13 Apr 2026 16:02:24 UTC (30 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)