When Are Two Probability Densities Fourier Compatible? The One-Dimensional Pauli Problem

The Pauli problem asks whether a quantum state is uniquely determined by its position and momentum probability densities. In this talk, we address a logically prior question: given two probability densities, can they arise at all as the squared moduli of a normalized Fourier pair?

The Heisenberg uncertainty relation provides a well-known necessary condition, but it is far from sufficient. We derive two pairs of stronger compatibility conditions. The first consists of Stam inequalities involving the Fisher information of the prescribed densities. The second follows from the Wiener–Khinchin theorem and relates the Fourier transform of each density to the autocorrelation of the square root of the other. By analyzing their behavior near the origin, we show that the Wiener–Khinchin inequalities imply the corresponding Stam inequalities, establishing a hierarchy of compatibility criteria.

Examples based on harmonic-oscillator densities illustrate the different levels of this hierarchy. We also show that the Wiener–Khinchin conditions can detect incompatibility that remains invisible to both the Heisenberg and Stam inequalities. The results provide practical tests that can be applied before attempting phase retrieval and suggest broader connections between the Pauli problem, Fourier analysis, information-theoretic inequalities, and autocorrelation methods.