Combinatorics Seminar - Jean-Philippe Labbé

16:15-18:00

Speaker: Jean-Philippe Labbé

Title: Lineup polytopes and applications in quantum physics

Abstract: 

The set of all possible spectra of 1-reduced density operators for systems of N
particles on a d-dimensional Hilbert space is a polytope called hypersimplex. If
the spectrum of the original density operators is fixed, the set of spectra (ordered
decreasingly) of 1-reduced density operators is also a polytope. A theoretical
description of this polytope using inequalities was provided by Klyachko in the
early 2000’s.
Adapting and enhancing tools from discrete geometry and combinatorics (symmetric
polytopes, sweep polytopes, and the Gale order), we obtained such necessary
inequalities explicitly, that are furthermore valid for arbitrarily large N and d.
These may therefore be interpreted as generalizations of Pauli's exclusion principle
for fermions. In particular, this approach leads to a new class of polytopes called
lineup polytopes.

This is joint work with physicists Julia Liebert, Christian Schilling and mathematicians
Eva Philippe, Federico Castillo and Arnau Padrol.