Precision Bounds on Continuous-Variable State Tomography Using Classical Shadows

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, and photon-parity protocols. To reach a desired precision on the classical shadow of an $N$-photon density matrix with high probability, we show that homodyne detection requires order $O\left(N^{4+1/3}\right)$ measurements in the worst case, whereas photon-number-resolving and photon-parity detection require $O\left(N^4\right)$ 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 analyses of homodyne tomography match closely with our theoretical predictions. We extend our single-mode results to an efficient construction of multimode shadows based on local measurements.