Derivation of the Loop Hafnian Generating Function for Arbitrary Symmetric Matrices via Gaussian Integration

TL;DR

This paper proves that the loop hafnian generating function, previously derived using quantum optics for specific matrices, is valid for all symmetric matrices using Gaussian integration, broadening its applicability.

Contribution

It provides a purely mathematical proof that extends the validity of the loop hafnian generating function to all symmetric matrices, removing quantum-specific assumptions.

Findings

The generating function applies to all symmetric matrices.

Gaussian integration suffices for the proof.

No quantum state properties are needed for the derivation.

Abstract

This short note shows that the recently proposed generating function for loop hafnians -- originally derived using quantum-optical methods for a restricted class of matrices -- is in fact valid for arbitrary symmetric matrices. The proof relies solely on Gaussian integration and does not assume any additional properties inherited from the covariance matrices of quantum Gaussian states.