On distances among Slater Determinant States and Determinantal Point Processes

TL;DR

The paper develops a quantitative bridge between quantum fermionic Slater determinant states and classical determinantal point processes, by linking quantum state distances (trace distance and quantum $W_1$) to classical transport distances (TV and $W_1$). It derives bounds on the quantum $W_1$ distance between Slater determinants in terms of unitary overlaps and demonstrates a monotonicity property for $k$-particle reduced density matrices. These quantum-to-classical bounds are then transported to determinantal point processes with projection kernels, yielding explicit TV and $W_\#$ distance bounds between laws via a Slater-based coupling. The results advance understanding of how quantum geometry constrains classical determinantal models and point to future work on sharpness, alternative Wasserstein formulations, and computational methods in quantum chemistry and related areas.

Abstract

Determinantal processes provide mathematical modeling of repulsion among points. In quantum mechanics, Slater determinant states generate such processes, reflecting Fermionic behavior. This note exploits the connections between the former and the latter structures by establishing quantitative bounds in terms of trace/total variation and Wasserstein distances.

Theorems & Definitions (13)

• Example 2.1
• Remark 2.2
• Remark 2.3
• Theorem 3.1
• proof
• Example 3.2
• Proposition 4.1
• Lemma 4.2
• proof
• proof : Proof of \ref{['prop:determinant-distance']}