aiqbench analysis
This research establishes a local differential hierarchy for the Husimi representation, proving that quantum state positivity and operator rank can be determined by conditions at single points in phase space rather than global constraints. By linking these local derivatives to the nonnegativity of determinants in a displaced Fock basis, the study reinforces phase-space representations as a robust geometric framework for distinguishing quantum features from classical ones.
Differential hierarchy of the Husimi representation
arXiv:2609.29388v1 Announce Type: new 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.
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.
4+8 pages. Comments welcome
arXiv:2609.29388 [quant-ph]
(or arXiv:2609.29388v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2609.29388 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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