  Quantum gas microscopy has developed into a powerful tool to explore strongly
correlated quantum systems. However, discerning phases with topological or
off-diagonal long range order requires the ability to extract these
correlations from site-resolved measurements. Here, we show that a multi-scale
complexity measure can pinpoint the transition to and from the bond ordered
wave phase of the one-dimensional extended Hubbard model with an off-diagonal
order parameter, sandwiched between diagonal charge and spin density wave
phases, using only diagonal descriptors. We study the model directly in the
thermodynamic limit using the recently developed variational uniform matrix
product states algorithm, and draw our samples from degenerate ground states
related by global spin rotations, emulating the projective measurements that
are accessible in experiments. Our results will have important implications for
the study of exotic phases using optical lattice experiments.
