Projections of Orbital Measures and Quantum Marginal Problems

TL;DR

This paper investigates the projections of random elements in Lie group orbits, deriving formulas for their densities and exploring implications for quantum marginal problems and representation theory.

Contribution

It introduces integral formulas for densities of projections in Lie group orbits and connects these to quantum information and multiplicity problems.

Findings

Derived explicit integral formulas for probability densities.

Established properties of these densities and their geometric significance.

Applied results to quantum marginal problems and representation theory.

Abstract

This paper studies projections of uniform random elements of (co)adjoint orbits of compact Lie groups. Such projections generalize several widely studied ensembles in random matrix theory, including the randomized Horn's problem, the randomized Schur's problem, and the orbital corners process. In this general setting, we prove integral formulae for the probability densities, establish some properties of the densities, and discuss connections to multiplicity problems in representation theory as well as to known results in the symplectic geometry literature. As applications, we show a number of results on marginal problems in quantum information theory and also prove an integral formula for restriction multiplicities.