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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(N4+1/3) measurements in the worst case, whereas photon-number-resolving and photon-parity detection require O(N4) 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. 

Techniques to control the spectro-temporal properties of quantum states of light at ultrafast time scales are crucial for numerous applications in quantum information science. In this work, we report an all-optical time lens for quantum signals based on Bragg-scattering four-wave mixing with picosecond resolution. Our system achieves a temporal magnification factor of 158 with single-photon level inputs, which is sufficient to overcome the intrinsic timing jitter of superconducting nanowire single-photon detectors. We demonstrate discrimination of two terahertz-bandwidth, single-photon-level pulses with 2.1 ps resolution (electronic jitter corrected resolution of 1.25 ps). We draw on elegant tools from Fourier optics to further show that the time-lens framework can be extended to perform complex unitary spectro-temporal transformations by imparting optimized temporal and spectral phase profiles to the input waveforms. Using numerical optimization techniques, we show that a four-stage transformation can realize an efficient temporal mode sorter that demultiplexes 10 Hermite–Gaussian (HG) modes. Our time-lens-based framework represents a new toolkit for arbitrary spectro-temporal processing of single photons, with applications in temporal mode quantum processing, high-dimensional quantum key distribution, temporal mode matching for quantum networks, and quantum-enhanced sensing with time-frequency entangled states.