Classical shadow tomography for continuous variables quantum systems

Understand

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 $M_n$.