Upper and lower bounds for few-copy Gaussian tomography, and the absence of a fermionic warm regime

arXiv:2611.07271v1 Announce Type: new Abstract: Chen, Mele, Mele and Preskill proved a crossover for bosonic Gaussian tomography: cold states need $\Theta(m^3/\epsilon^2)$ copies under single-copy or few-copy measurements, warm ones $\Theta(m^2/\epsilon^2)$. The author proved $\Omega(m^3/\epsilon^2)$ for adaptive single-copy tomography of mixed fermionic Gaussian states. This note shows the crossover has no fermionic counterpart and extends the fermionic bound to blocks of $k$ copies. There is no warm regime: on the quadratic Gibbs balls $\beta\|h\|_F\le\beta_0$ of unknown Hamiltonian $h$, the single-copy rate is $\Theta(m^3/\epsilon^2)$ for every fixed $\beta_0>0$, at accuracies that resolve the ball, with a lower-bound constant independent of $\beta_0$.