On the complex zeros of the wavefunction

TL;DR

This paper investigates the zeros of bosonic wavefunctions, revealing their complex-analytic structure and linking non-Gaussianity in quantum states to the distribution of these zeros, with implications for quantum state characterization.

Contribution

It introduces a novel complex analysis approach to bosonic wavefunctions, extending them holomorphically and characterizing Gaussian dynamics via wavefunction zeros.

Findings

Most bosonic wavefunctions can be extended to holomorphic functions.

A version of Hudson's theorem for bosonic wavefunctions is proved.

Non-Gaussianity can be detected by measuring a single quadrature.

Abstract

The Schr\"odinger wavefunction is ubiquitous in quantum mechanics, quantum chemistry, and bosonic quantum information theory. Its zero-set for fermionic systems is well-studied and central for determining chemical properties, yet for bosonic systems the zero-set is less understood, especially in the context of characterizing non-classicality. Here we study the zeros of such wavefunctions and give them a novel information-theoretic interpretation. Our main technical result is showing that the wavefunction of most bosonic quantum systems can be extended to a holomorphic function over the complex plane, allowing the application of powerful techniques from complex analysis. As a consequence, we prove a version of Hudson's theorem for the wavefunction and characterize Gaussian dynamics as classical motion of the wavefunction zeros. Our findings suggest that the non-Gaussianity of quantum optical states can be detected by measuring a single quadrature of the electromagnetic field, which we demonstrate in a companion paper [arXiv:2507.23005]. More generally, our results show that the non-Gaussian features of bosonic quantum systems are encoded in the zeros of their wavefunction.