Spectral-angular parametrization of open qudit dynamics: gap coordinates, Cartan structure, and GKLS decoupling

Abstract

Every $n$-level density matrix can be diagonalized as $\rho=U\,\mathrm{diag}(p_1,\ldots,p_n)U^\dagger$, splitting it into an eigenvalue vector $(p_1,\ldots,p_n)$ and a unitary eigenbasis $U$. We introduce a new coordinate system on the eigenvalue vector alone: the \emph{gap coordinates} $r_a:=p_a-p_{a+1}\ge0$, $a=1,\ldots,n-1$. We show that $(r_1,\ldots,r_{n-1})$ are precisely the simple-root coordinates of the eigenvalue vector in the Cartan subalgebra of $A_{n-1}=\mathfrak{sl}(n)$, so that the spectral diagonal expands in the fundamental-coweight basis, $\mathsf D(\mathbf r)=\sum_a r_a\,\omega_a^\vee$, and the $n-1$ eigenvalue-ordering inequalities collapse into the single linear constraint $\sum_a a\,r_a\le1$ defining a weighted simplex $R_{n-1}$. This identification yields closed-form expressions for the Fisher--Rao and Bures metrics at the maximally mixed state in terms of the inverse Cartan matrix of $A_{n-1}$, and for a piecewise-linear alternative to the standard definition of purity. On the angular side we give an explicit Tilma--Sudarshan-type coordinatization of the flag manifold $F_n\simeq\mathrm{SU}(n)/\mathbb T^{n-1}$ and a global coherent-state resolution of the identity on $F_n$, enabling an $\mathrm{SU}(n)$-covariant integral quantization of the flag variety. Under GKLS dynamics, the pair (gap coordinates, flag angles) exhibits a partial decoupling: the eigenvalue gaps evolve under dissipation alone, while the flag angles are driven by both the Hamiltonian and the dissipator, which we derive directly in these coordinates, including an explicit illustration on the real qutrit sector.