Differential hierarchy of the Husimi representation
Abstract
Phase-space representations reveal the geometry of quantum mechanics and distinguish quantum features from classical ones. However, the positivity of a density operator is among the properties of quantum states that phase-space representations capture least directly: deciding whether a phase-space function describes a physical quantum state has so far required global conditions coupling distant points in phase space. Here we show that these conditions can in fact be made local, by considering the Husimi representation in phase space. We establish a complete set of conditions which captures when a phase-space function f is the Husimi function of a quantum state, with each condition involving only the derivatives of the function f evaluated at a single phase-space point. We further show that each condition has an equivalent algebraic characterization, which takes the form of the nonnegativity of a determinant of the underlying operator in a displaced Fock basis. As a result, these conditions obey a recurrence relation organizing them into a differential hierarchy, which terminates based on the rank of the operator. Our results thus show that operator rank and positivity are local properties in phase space, reinforcing the role of phase-space representations as a geometric description of quantum mechanics.
Related
- From a Hierarchy of Stochastic Differential Equations to a Hierarchy of Generalized Beta Distributions
- Canonical differential equations for Feynman integrals from 𝒜-hypergeometric systems in the Schwinger representation
- Irreversibility of many-body harmonic oscillators characterized by generalized Husimi's adiabaticity parameter
- Comment on "The polarization hierarchy for polynomial optimization over convex bodies, with applications to nonnegative matrix rank"
- Unfolded Schwinger-Dyson hierarchy for vertex functions
BibTeX
@misc{herstraeten2026differential,
title = {{Differential hierarchy of the Husimi representation}},
author = {Zacharie Van Herstraeten and Ulysse Chabaud},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29388},
eprint = {2609.29388},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29388},
}
RIS
TY - UNPB
TI - Differential hierarchy of the Husimi representation
AU - Zacharie Van Herstraeten
AU - Ulysse Chabaud
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29388
UR - https://arxiv.org/abs/2609.29388
AB - Phase-space representations reveal the geometry of quantum mechanics and distinguish quantum features from classical ones. However, the positivity of a density operator is among the properties of quantum states that phase-space representations capture least directly: deciding whether a phase-space function describes a physical quantum state has so far required global conditions coupling distant points in phase space. Here we show that these conditions can in fact be made local, by considering the Husimi representation in phase space. We establish a complete set of conditions which captures when a phase-space function f is the Husimi function of a quantum state, with each condition involving only the derivatives of the function f evaluated at a single phase-space point. We further show that each condition has an equivalent algebraic characterization, which takes the form of the nonnegativity of a determinant of the underlying operator in a displaced Fock basis. As a result, these conditions obey a recurrence relation organizing them into a differential hierarchy, which terminates based on the rank of the operator. Our results thus show that operator rank and positivity are local properties in phase space, reinforcing the role of phase-space representations as a geometric description of quantum mechanics.
ER -