Entangled measurements are necessary for optimal tomography of mixed fermionic Gaussian states and of bosonic Gaussian states near the vacuum

We prove that learning an unknown mixed fermionic Gaussian state on $m$ modes to trace distance $ε$ requires $Ω(m^3/ε^2)$ copies when measurements act on one copy at a time, even with arbitrary POVMs, fresh ancillas and classical adaptivity, but without quantum memory between copies. The bound holds