  Quantum simulations with ultra-cold atoms in optical lattices open up an
exciting path towards understanding strongly interacting quantum systems. Atom
gas microscopes are crucial for this as they offer single-site density
resolution, unparalleled in other quantum many-body systems. However, currently
a direct measurement of local coherent currents is out of reach. In this work,
we show how to achieve that by measuring densities that are altered in response
to quenches to non-interacting dynamics, e.g., after tilting the optical
lattice. For this, we establish a data analysis method solving the closed set
of equations relating tunnelling currents and atom number dynamics, allowing to
reliably recover the full covariance matrix, including off-diagonal terms
representing coherent currents. The signal processing builds upon semi-definite
optimization, providing bona fide covariance matrices optimally matching the
observed data. We demonstrate how the obtained information about non-commuting
observables allows to lower bound entanglement at finite temperature which
opens up the possibility to study quantum correlations in quantum simulations
going beyond classical capabilities.
