Sample complexity of learning fermionic and bosonic Gaussian states with single-copy and entangled measurements

arXiv:2611.13428v1 Announce Type: new Abstract: With single-copy measurements, every mixed fermionic Gaussian state on $m$ modes, pure modes included, is learned to trace distance $\epsilon$ from $\Theta(m^3/\epsilon^2)$ copies, with no logarithmic factor, against the known $\Theta(m^2/\epsilon^2)$ for collective measurements. We prove the upper bound, resolving Conjecture 10.1 of arXiv:2609.23189 (lower bound for $\epsilon\le1/3600$), and fill most open entries of the sample-complexity table of fermionic and bosonic Gaussian states, for single-copy, $k$-copy and collective measurements. The single-copy measurement's noise is matched to the metric at every occupation scale.