Quantum Marginal Problem and its Physical Relevance

TL;DR

This paper explores the mathematical foundations and physical implications of generalized Pauli constraints (GPC) in fermionic systems, revealing that quasipinning phenomena are prevalent in interacting fermions and may have significant physical relevance.

Contribution

It develops an accessible understanding of Klyachko's solution to the quantum marginal problem and investigates the physical relevance of GPC, especially quasipinning, in concrete fermionic systems.

Findings

Natural occupation numbers exhibit strong quasipinning in interacting fermions.

Pinning of GPC is considered unnatural, but quasipinning is common.

Quasipinning is linked to the structure of low-energy eigenstates.

Abstract

The Pauli exclusion principle as constraint on fermionic occupation numbers is a consequence of the much deeper fermionic exchange statistics. Just recently, it was shown by Klyachko that this antisymmetry of fermionic wave functions leads to further restrictions on natural occupation numbers. These so-called generalized Pauli constraints (GPC) significantly strengthen Pauli's exclusion principle. Our first goal is to develop an understanding of the mathematical concepts behind Klyachko's work, in the context of quantum marginal problems. Afterwards, we explore the physical relevance of GPC and study concrete physical systems from that new viewpoint. In the first part of this thesis we review Klyachko's solution of the univariate quantum marginal problem. In particular we break his abstract derivation based on algebraic topology down to a more elementary level and reveal the geometrical picture behind it. The second part explores the possible physical relevance of GPC. We review the effect of pinning, i.e. the saturation of some GPC by given natural occupation numbers and explain its consequences. Although this effect would be quite spectacular we argue that pinning is unnatural. Instead, we conjecture the effect of quasipinning, defined by occupation numbers close to (but not exactly on) the boundary of the allowed region. In the third part we study concrete fermionic quantum systems from the new viewpoint of GPC. In particular, we compute the natural occupation numbers for the ground state of a family of interacting fermions in a harmonic potential. Intriguingly, we find that the occupation numbers are strongly quasipinned, even up to medium interaction strengths. We identify this as an effect of the lowest few energy eigenstates, which provides first insights into the mechanism behind quasipinning.