Multimode and Experimental Continuous Variable Shadow Tomography

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. Earlier, we extended this framework to continuous-variable (CV) quantum systems, such as optical modes and harmonic oscillators. Constraining the occupation number, such as the photon-number for optical systems, to a maximum of N, we provided rigorous bounds on the sample complexity of single-mode states and applied our framework to existing optical tomography methods.

We now extend this framework to multimode CV systems which consist of multiple modes of light possibly entangled with each other, analogous to multi-qubit systems in the discrete-variable scenario. Restricting each mode to a maximum occupational number N, we show that the number of samples required to estimate a multimode state to a desired level of accuracy w.h.p. scales exponentially with the number of modes. However, we also show that this method is efficient for reconstructing all local reduced density matrices of constant size. We support our findings with numerical simulations.

We also benchmark our single-mode analysis techniques on experimental homodyne data measured on photon subtracted coherent states with multiple detection methods. This experimental data has shown a highly favorable scaling of the number of measurements required.