Optimal constant in fermionic Gaussianity testing copy complexity

Determine whether collective tests more sophisticated than the Bell-like tests can improve the precise constant in the copy complexity required to distinguish pure fermionic Gaussian states from pure states at least ε-far from the Gaussian manifold.

Background

The paper proves that fermionic Gaussianity can be tested with Θ(ε⁻² log(1/δ)) copies, establishing optimal asymptotic dependence on the distance and failure-probability parameters and removing dependence on the number of fermionic modes. The result is obtained for the Bell sampling procedure and therefore determines the scaling only up to multiplicative constants.

The authors explicitly leave unresolved whether more involved collective measurements can achieve a smaller constant than the Bell-like tests. This concerns the exact finite-constant optimality of the testing procedure, not its asymptotic scaling.