Given a random quantum state of multiple distinguishable or indistinguishable
particles, we provide an effective method, rooted in symplectic geometry, to
compute the joint probability distribution of the eigenvalues of its one-body
reduced density matrices. As a corollary, by taking the distribution's support,
which is a convex moment polytope, we recover a complete solution to the
one-body quantum marginal problem. We obtain the probability distribution by
reducing to the corresponding distribution of diagonal entries (i.e., to the
quantitative version of a classical marginal problem), which is then determined
algorithmically. This reduction applies more generally to symplectic geometry,
relating invariant measures for the coadjoint action of a compact Lie group to
their projections onto a Cartan subalgebra, and can also be quantized to
provide an efficient algorithm for computing bounded height Kronecker and
plethysm coefficients.Comment: 51 pages, 7 figure
We study the generalized Galois numbers which count flags of length r in N-dimensional vector spaces over finite fields. We prove that the coefficients of those polynomials are asymptotically Gaussian normally distributed as N becomes large. Furthermore, we interpret the generalized Galois numbers as weighted inversion statistics on the descent classes of the symmetric group on N elements and identify their asymptotic limit as the Mahonian inversion statistic when r approaches ∞. Finally, we apply our statements to derive further statistical aspects of generalized Rogers-Szegő polynomials, reinterpret the asymptotic behavior of linear q-ary codes and characters of the symmetric group acting on subspaces over finite fields, and discuss implications for affine Demazure modules and joint probability generating functions of descent-inversion statistics.
We compute the expected degree of a randomly chosen element in a basis of weight vectors in the Demazure module Vw(Λ) of sl 2 . We obtain en passant a new proof of Sanderson's dimension formula for these Demazure modules.
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