# Quantum marginals from pure doubly excited states

## TL;DR

This paper introduces a new geometric approach to characterize the spectra of one-particle reduced density matrices in pure quantum states, providing bounds and exact descriptions for specific quantum systems using symplectic geometry.

## Contribution

It develops inner and outer polytope bounds for quantum spectra and identifies systems where these bounds coincide, using advanced mathematical tools from symplectic geometry and Lie group theory.

## Key findings

- Outer bound is sharp for the entire quantum system polytope.
- Inner bound derived from doubly excited states matches the outer bound for specific systems.
- Identifies classes of quantum systems where bounds coincide, fully characterizing their spectra.

## Abstract

The possible spectra of one-particle reduced density matrices that are compatible with a pure multipartite quantum system of finite dimension form a convex polytope. We introduce a new construction of inner- and outer-bounding polytopes that constrain the polytope for the entire quantum system. The outer bound is sharp. The inner polytope stems only from doubly excited states. We find all quantum systems, where the bounds coincide giving the entire polytope. We show, that those systems are: i) any system of two particles ii) $L$ qubits, iii) three fermions on $N\leq 7$ levels, iv) any number of bosons on any number of levels and v) fermionic Fock space on $N\leq 5$ levels. The methods we use come from symplectic geometry and representation theory of compact Lie groups. In particular, we study the images of proper momentum maps, where our method describes momentum images for all representations that are spherical.