Shadow tomography on general measurement frames

Abstract

We provide a new perspective on shadow tomography by demonstrating its deep connections with the general theory of measurement frames. By showing that the formalism of measurement frames offers a natural framework for shadow tomography -- in which ``classical shadows'' correspond to unbiased estimators derived from a suitable dual frame associated with the given measurement -- we highlight the intrinsic connection between standard state tomography and shadow tomography. Such perspective allows us to examine the interplay between measurements, reconstructed observables, and the estimators used to process measurement outcomes, while paving the way to assess the influence of the input state and the dimension of the underlying space on estimation errors. Our approach generalizes the method described in [H.-Y. Huang {\it et al.}, Nat. Phys. 16, 1050 (2020)], whose results are recovered in the special case of covariant measurement frames. As an application, we demonstrate that a sought-after target of shadow tomography can be achieved for the entire class of tight rank-1 measurement frames -- namely, that it is possible to accurately estimate a finite set of generic rank-1 bounded observables while avoiding the growth of the number of the required samples with the state dimension.

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• Optimising quantum tomography via shadow inversion
• Quantum Extreme Learning of molecular potential energy surfaces and force fields
• Quantum reservoir computing for photonic entanglement witnessing
• Qudit Shadow Estimation Based on the Clifford Group and the Power of a Single Magic Gate
• Completeness Stability of Quantum Measurements
• Low variance estimations of many observables with tensor networks and informationally-complete measurements
• Chasing shadows with Gottesman-Kitaev-Preskill codes
• Classification of quantum states of light using random measurements through a multimode fiber
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• Quantum Advantage via Efficient Post-processing on Qudit Classical Shadow tomography
• Improving shadow estimation with locally-optimal dual frames
• Reducing Complexity of Shadow Process Tomography with Generalized Measurements