Spectral-angular parametrization of open qudit dynamics

Abstract

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 $\mathbf{r}$.