  Estimating physical properties of quantum states from measurements is one of
the most fundamental tasks in quantum science. In this work, we identify
conditions on states under which it is possible to infer the expectation values
of all quasi-local observables of a state from a number of copies that scales
polylogarithmically with the system's size and polynomially on the locality of
the target observables. We show that this constitutes a provable exponential
improvement in the number of copies over state-of-the-art tomography protocols.
We achieve our results by combining the maximum entropy method with tools from
the emerging fields of classical shadows and quantum optimal transport. The
latter allows us to fine-tune the error made in estimating the expectation
value of an observable in terms of how local it is and how well we approximate
the expectation value of a fixed set of few-body observables. We conjecture
that our condition holds for all states exhibiting some form of decay of
correlations and establish it for several subsets thereof. These include widely
studied classes of states such as one-dimensional thermal and high-temperature
Gibbs states of local commuting Hamiltonians on arbitrary hypergraphs or
outputs of shallow circuits. Moreover, we show improvements of the maximum
entropy method beyond the sample complexity that are of independent interest.
These include identifying regimes in which it is possible to perform the
postprocessing efficiently as well as novel bounds on the condition number of
covariance matrices of many-body states.
