Shadow tomography on general measurement frames

TL;DR

This paper connects shadow tomography with measurement frames, providing a generalized framework that improves understanding of estimation errors and enables efficient estimation of observables without increasing sample complexity with system dimension.

Contribution

It introduces a measurement frame perspective on shadow tomography, generalizing previous methods and enabling efficient estimation of observables across a broad class of measurement frames.

Findings

Shadow tomography can be viewed through the lens of measurement frames.

The approach generalizes existing shadow tomography methods.

Efficient estimation of certain observables is possible without sample size growth.

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.