  High-dimensional entanglement has been identified as an important resource in
quantum information processing, and also as a main obstacle for simulating
quantum systems. Its certification is often difficult, and most widely used
methods for experiments are based on fidelity measurements with respect to
highly entangled states. Here, instead, we consider covariances of collective
observables, as in the well-known Covariance Matrix Criterion (CMC)[1] and
present a generalization of the CMC for determining the Schmidt number of a
bipartite system. This is potentially particularly advantageous in many-body
systems, such as cold atoms, where the set of practical measurements is very
limited and only variances of collective operators can typically be estimated.
To show the practical relevance of our results, we derive simpler
Schmidt-number criteria that require similar information as the fidelity-based
witnesses, yet can detect a wider set of states. We also consider paradigmatic
criteria based on spin covariances, which would be very helpful for
experimental detection of high-dimensional entanglement in cold atom systems.
We conclude by discussing the applicability of our results to a multiparticle
ensemble and some open questions for future work.
