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(m3/ε2) copies under single-copy or few-copy measurements, warm ones Θ(m2/ε2). The author proved Ω(m3/ε2) 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≤β0 of unknown Hamiltonian h, the single-copy rate is Θ(m3/ε2) for every fixed β0>0, at accuracies that resolve the ball, with a lower-bound constant independent of β0. At the maximally mixed state a replicated-parameter Fisher budget gives N>m3/(8,553,600 k ε2) for blocks of at most k copies and all ε≤1/3600; an exact formula for the maximal block Fisher information, via spinor tensor powers of so(2m), shows that the budget mt2 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(m3/(ε2min{√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.
Related
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BibTeX
@misc{rubin2026upper,
title = {{Upper and lower bounds for few-copy Gaussian tomography, and the absence of a fermionic warm regime}},
author = {Ron Rubin},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07271},
eprint = {2611.07271},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07271},
}
RIS
TY - UNPB
TI - Upper and lower bounds for few-copy Gaussian tomography, and the absence of a fermionic warm regime
AU - Ron Rubin
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07271
UR - https://arxiv.org/abs/2611.07271
AB - 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≤β_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≤1/3600; an exact formula for the maximal block Fisher information, via spinor tensor powers of 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^2min{√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.
ER -