Eigenvalue Distributions of Reduced Density Matrices

TL;DR

This paper introduces a symplectic geometry-based method to compute eigenvalue distributions of reduced density matrices in quantum states, solving the quantum marginal problem and connecting to classical marginal problems and representation theory.

Contribution

It provides a novel, effective approach rooted in symplectic geometry to determine eigenvalue distributions and solve the quantum marginal problem for complex quantum systems.

Findings

Derived the joint eigenvalue distribution for reduced density matrices.

Solved the quantum marginal problem using convex polytopes.

Connected eigenvalue distributions to classical marginal problems and representation theory.

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 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.