Before revision:  Thus, our result has several direct applications, including polynomial-time algorithms 1)  to sample from  matrix Langevin distributions studied in statistics, 2)  to sample from continuous maximum entropy distributions with applications to quantum inference , and 3)  for differentially private low-rank approximation. The key obstacle in sampling from an HCIZ density is that the domain is an algebraic manifold and the entries of the sample matrix are highly correlated. To overcome this, we consider a mapping that sends each Hermitian matrix X to a natural ordering of the eigenvalues of all leading principal minors of X. The image of each \mathrm{U(n)-orbit under this map is a convex polytope, and the mapping reveals a recursive structure of \mathrm{U}(n)-orbits.
After revision:  Thus, our result has several direct applications, including polynomial-time algorithms 1) to sample from matrix Langevin distributions studied in statistics, 2) to sample from continuous maximum entropy distributions with applications to quantum inference , and 3) for differentially private (n)-orbit under this map is a convex polytope, and the mapping reveals a recursive structure of \mathrm{U}(n)-orbits.