Eigenvalue Distributions of Reduced Density Matrices
Abstract
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 onebody 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 onebody 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.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 November 2014
 DOI:
 10.1007/s0022001421444
 arXiv:
 arXiv:1204.0741
 Bibcode:
 2014CMaPh.332....1C
 Keywords:

 Quantum Physics;
 Mathematical Physics;
 Mathematics  Algebraic Geometry;
 Mathematics  Symplectic Geometry
 EPrint:
 51 pages, 7 figures