  The fact that quantum mechanics predicts stronger correlations than classical
physics is an essential cornerstone of quantum information processing. Indeed,
these quantum correlations are a valuable resource for various tasks, such as
quantum key distribution or quantum teleportation, but characterizing these
correlations in an experimental setting is a formidable task, especially in
scenarios where no shared reference frames are available. By definition,
quantum correlations are reference-frame independent, i.e., invariant under
local transformations; this physically motivated invariance implies, however, a
dedicated mathematical structure and, therefore, constitutes a roadblock for an
efficient analysis of these correlations in experiments. Here we provide a
method to directly measure any locally invariant property of quantum states
using locally randomized measurements, and we present a detailed toolbox to
analyze these correlations for two quantum bits. We implement these methods
experimentally using pairs of entangled photons, characterizing their
usefulness for quantum teleportation and their potential to display quantum
nonlocality in its simplest form. Our results can be applied to various quantum
computing platforms, allowing simple analysis of correlations between arbitrary
distant qubits in the architecture.
