Shadow Tomography on General Measurement Frames

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 a perspective allows us to examine the interplay between measurements, reconstructed observables, and the estimators used to process measurement outcomes, while paving the way to assessing the influence of the input state and the dimension of the underlying space on estimation errors. Our approach generalizes the method described by Huang et al. [H.-Y. Huang 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.