Full joint distribution of qubit and qutrit marginal spectra

Derive the full joint distribution of the qubit Bloch radius and the largest and smallest eigenvalues of the qutrit marginal for Haar-random states on a fixed-spectrum qubit-qutrit unitary orbit.

Background

The paper derives separate one-marginal laws for the qubit Bloch radius and for the largest and smallest eigenvalues of the qutrit marginal in the qubit-qutrit case. Because the qutrit trace is fixed, its largest and smallest eigenvalues determine the full qutrit spectrum, but the paper does not derive their joint distribution with the qubit Bloch radius.

The unresolved distribution is expected to involve a multivariate spline supported on the full qubit-qutrit marginal polytope. This would extend the paper’s explicit two-qubit joint-density results to the coupled three-variable spectral data of a qubit-qutrit system.