Classical shadow tomography for continuous variables quantum systems

In this article we develop a continuous variable (CV) shadow tomography scheme with wide ranging applications in quantum optics. Our work is motivated by the increasing experimental and technological relevance of CV systems in quantum information, quantum communication, quantum sensing, quantum simulations, quantum computing and error correction. We introduce two experimentally realisable schemes for obtaining classical shadows of CV (possibly non-Gaussian) quantum states using only randomised Gaussian unitaries and easily implementable Gaussian measurements such as homodyne and heterodyne detection. For both schemes, we show that (formula presented) samples of an unknown m-mode state ρ suffice to learn the expected value of any r -local polynomial in the canonical observables of degree α , both with high probability 1-δ and accuracy ϵ , as long as the state ρ has moments of order n>α bounded by Mn. By simultaneously truncating states and operators in energy and phase space, we are abl...