Testing bosonic and fermionic Gaussian states with a constant number of copies
Jop Briët, Davi Castro-Silva, Michael Walter, Freek Witteveen
quant-ph
Oct 5, 2026 · v2
TL;DR
The authors provide a Lean formalization of their Gaussianity-testing results in a public GitHub repository, noted in a footnote.
Abstract
Bosonic and fermionic Gaussian quantum states play a central role in quantum physics and quantum information processing. We study the problem of testing whether a given pure state is Gaussian or is bounded away from the set of pure Gaussian states. Earlier work left open whether a fixed number of copies could suffice even as the number of modes (system size) grows. We show that it does: the number of copies needed depends only on the desired accuracy and confidence. Our tests exploit the symmetries of identical copies of Gaussian states; their analysis combines a gradient flow argument with a spectral gap studied previously for the Kac random walk on the sphere.
Problem
It was open whether pure bosonic and fermionic Gaussian states can be tested with a number of copies independent of the number of modes. Earlier bounds grew with system size.
Approach
Tests are built from symmetries of identical copies of Gaussian states, such as projection onto SO(2)-invariant subspaces of two copies. Robustness comes from a gradient flow of the acceptance probability, with the gradient bounded by the spectral gap of a three-copy symmetrized operator. That gap is obtained from the known Kac random walk spectral gap (5/12 for t=3). A Lean formalization is provided.
Results
Bosonic and fermionic Gaussian states can be tested with one-sided error using O(ε^-2 log(1/δ)) copies, independent of the number of modes. Similar tests are given for mean-zero bosonic Gaussian states and for coherent states.
