  Recent progress in quantum optics has led to setups that are able to prepare
high-dimensional quantum states for quantum information processing tasks. As
such, it is of importance to benchmark the states generated by these setups in
terms of their quantum mechanical properties, such as their Schmidt numbers,
i.e., the number of entangled degrees of freedom. In this paper, we develop an
iterative algorithm that finds Schmidt number witnesses tailored to the
measurements available in specific experimental setups. We then apply the
algorithm to find a witness that requires the measurement of a number of
density matrix elements that scales linearly with the local dimension of the
system. As a concrete example, we apply our construction method to an
implementation with photonic temporal modes.
