[Paper Review] Quantum Marginal Problem and its Physical Relevance

Christian Schilling|arXiv (Cornell University)|Jan 1, 2014

Quantum many-body systemsPhysics and Astronomy38 references21 citations

TL;DR

This dissertation develops a geometric framework for the quantum marginal problem, revealing that generalized Pauli constraints—derived from antisymmetry of N-fermion wavefunctions—impose deeper restrictions on natural occupation numbers (NONs). The author introduces 'quasi-pinning' as a physically relevant phenomenon where NONs are near, but not exactly on, the boundary of allowed regions, and demonstrates strong quasi-pinning in interacting fermions in harmonic traps and exact pinning in symmetric Hubbard models with three electrons on three sites.

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.

Motivation & Objective

• . To develop an accessible, geometric understanding of Klyachko's solution to the univariant quantum marginal problem using elementary algebraic geometry.
• . To investigate the physical relevance of generalized Pauli constraints, particularly the phenomenon of 'pinning' where occupation numbers lie exactly on the boundary of allowed regions.
• . To propose and formalize the concept of 'quasi-pinning'—occupation numbers near, but not on, the boundary—as a more natural and physically relevant phenomenon.
• . To apply the generalized Pauli constraints to concrete fermionic systems, including interacting fermions in a harmonic potential and the three-electron Hubbard model, to uncover physical mechanisms behind quasi-pinning.
• . To introduce a 'truncated pinning analysis' as a systematic tool for quantifying quasi-pinning in many-body quantum systems.

Proposed method

• . Uses algebraic topology and Schubert calculus to solve the univariant quantum marginal problem, reducing Klyachko’s abstract derivation to a geometric picture.
• . Employs generalized flag varieties and Grassmannians to describe the space of allowed one-body density matrices.
• . Applies homology and cohomology theory to characterize the boundary of the allowed region for occupation numbers.
• . Introduces the concept of 'quasi-pinning' by analyzing how occupation numbers approach, but do not reach, the boundary defined by generalized Pauli constraints.
• . Develops a 'truncated pinning analysis' to systematically quantify the degree of quasi-pinning in many-body states.
• . Computes natural occupation numbers for the ground state of interacting fermions in a harmonic trap using effective Hamiltonians and eigenvalue methods.

Experimental results

Research questions

• RQ1 . How can Klyachko’s abstract solution to the univariant quantum marginal problem be re-expressed in a more geometric and accessible way using elementary algebraic geometry?
• RQ2 . Is exact 'pinning'—where occupation numbers lie exactly on the boundary of the allowed region—physically plausible, or is it an artifact of symmetry?
• RQ3 . What is the physical significance of occupation numbers that are close to, but not exactly on, the boundary of the allowed region (i.e., 'quasi-pinning')?
• RQ4 . What many-body structures in fermionic systems are associated with quasi-pinning, and can they be systematically identified?
• RQ5 . How do symmetries in a model like the three-site Hubbard model influence the occurrence of exact pinning in its ground state?

Key findings

• . The natural occupation numbers of interacting fermions in a harmonic potential are strongly quasi-pinned even at medium interaction strengths, indicating a robust physical mechanism.
• . Quasi-pinning in the harmonic trap model is primarily driven by the lowest few energy eigenstates, suggesting a connection between low-energy structure and occupation number constraints.
• . Exact pinning in the three-electron Hubbard model occurs only when the system has high symmetry, indicating that such pinning is rare and structure-specific.
• . The generalized Pauli constraints for N=3, M=8 orbitals include 31 inequalities, with 14 of them being non-trivial and active in constraining the occupation numbers.
• . The concept of 'truncated pinning analysis' is introduced as a practical method to quantify quasi-pinning, enabling systematic study of its physical implications.
• . The study provides strong evidence that quasi-pinning corresponds to highly simplified, low-complexity N-fermion quantum states, suggesting deep physical relevance beyond exact pinning.