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

Abstract

Chen, Mele, Mele and Preskill proved a crossover for bosonic Gaussian tomography: cold states need $Θ(m^3/ε^2)$ copies under single-copy or few-copy measurements, warm ones $Θ(m^2/ε^2)$. The author proved $Ω(m^3/ε^2)$ 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 $β\|h\|_F\leβ_0$ of unknown Hamiltonian $h$, the single-copy rate is $Θ(m^3/ε^2)$ for every fixed $β_0>0$, at accuracies that resolve the ball, with a lower-bound constant independent of $β_0$. At the maximally mixed state a replicated-parameter Fisher budget gives $N>m^3/(8{,}553{,}600\,k\,ε^2)$ for blocks of at most $k$ copies and all $ε\le1/3600$; an exact formula for the maximal block Fisher information, via spinor tensor powers of $\mathfrak{so}(2m)$, shows that the budget $mt^2$ behind it is asymptotically attained for every fixed block size; whether $1/k$ is the true rate is open. Near the vacuum, block protocols need $Ω(m^3/(ε^2\min\{\sqrt k,m\}))$ copies, with no logarithm, via an exterior-power embedding of qudit tomography; parity superselection changes no rate, and number superselection costs nothing on number-conserving inputs. The embedding also carries upper bounds: a particle-number measurement followed by the $k$-copy qudit protocol of Nayak and Zhou attains the lower bound for every $k$ on number-conserving fermionic Gaussian states with bounded fugacity trace and on passive bosonic Gaussian states with bounded photon number, and a heterodyne stage that undoes displacement and squeezing extends this to all cold bosonic Gaussian states. This closes their $k$-entangled gap on the cold class; intermediate temperatures remain open.