Inverting Bosonic Gaussian Channels: One-Photon Tests and Sharp Tomography Rates

This preprint studies the reconstruction of arbitrary bosonic input states from the output of a known Gaussian channel. For families of channels with possibly varying input and output mode counts, a semidefinite condition characterizes uniform inverse continuity in trace distance under a total photon moment bound. The condition is equivalent to uniform separation in two types of vacuum and one-photon tests and gives a Gaussian receiver with a uniform logarithmic inverse bound, including in the presence of an untouched reference. For a fixed injective channel, the paper gives sharp logarithmic minimax rates for ordinary heterodyne tomography. The same rates hold for collective measurements when the channel is irreversible. For reversible channels, collective output tomography is equivalent to direct-input tomography, and a polynomial upper bound is given. The computational supplement reproduces 14,634 exact checks of finite identities, bounds, and examples in the manuscript.