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 $N=O\big(\operatorname{poly}\big(\frac{1}{\epsilon},\log\big(\frac{1}{\delta}\big),M_n^{r+\alpha},\log(m)\big)\big)$ samples of an unknown $m$-mode state $\rho$ suffice to learn the expected value of any $r$-local polynomial in the canonical observables of degree $\alpha$, both with high probability $1-\delta$ and accuracy $\epsilon$, as long as the state $\rho$ has moments of order $n>\alpha$ bounded by $M_n$. By simultaneously truncating states and operators in energy and phase space, we are able to overcome new mathematical challenges that arise due to the infinite-dimensionality of CV systems. We also provide a scheme to learn nonlinear functionals of the state, such as entropies over any small number of modes, by leveraging recent energy-constrained entropic continuity bounds. Finally, we provide numerical evidence of the efficiency of our protocols in the case of CV states of relevance in quantum information theory, including ground states of quadratic Hamiltonians of many-body systems and cat qubit states. We expect our scheme to provide good recovery in learning relevant states of 2D materials and photonic crystals.