Shadow Tomography of Continuous-Variable Quantum Systems

Shadow tomography is a framework for constructing succinct descriptions of quantum states, called classical shadows, with powerful methods to bound the estimators used. Classical shadows are well-studied in the discrete-variable case, which consists of states of several qubits.  Here, we extend this framework to continuous-variable quantum systems, such as optical modes and harmonic oscillators. We show how to adapt homodyne and photon number resolving (PNR) experimental methods from optical tomography to efficiently construct finite-dimensional classical shadows for an infinite-dimensional unknown state. We provide rigorous bounds on the variance of estimating density matrices from both of these experimental methods. We show that, to reach a desired precision on the classical shadow of an N-photon density matrix with a high probability, homodyne detection requires ~N⁵ measurements in the worst case, whereas PNR detection requires only ~N⁴ measurements in the worst case.