  Shadow tomography is a framework for constructing succinct descriptions of
quantum states using randomized measurement bases, called classical shadows,
with powerful methods to bound the estimators used. We recast existing
experimental protocols for continuous-variable quantum state tomography in the
classical-shadow framework, obtaining rigorous bounds on the number of
independent measurements needed for estimating density matrices from these
protocols. We analyze the efficiency of homodyne, heterodyne, photon number
resolving (PNR), and photon-parity protocols. To reach a desired precision on
the classical shadow of an $N$-photon density matrix with a high probability,
we show that homodyne detection requires an order $\mathcal{O}(N^{4+1/3})$
measurements in the worst case, whereas PNR and photon-parity detection require
$\mathcal{O}(N^4)$ measurements in the worst case (both up to logarithmic
corrections). We benchmark these results against numerical simulation as well
as experimental data from optical homodyne experiments. We find that numerical
and experimental homodyne tomography significantly outperforms our bounds,
exhibiting a more typical scaling of the number of measurements that is close
to linear in $N$. We extend our single-mode results to an efficient
construction of multimode shadows based on local measurements.

Quantum Physics