  Entanglement detection is one of the most conventional tasks in quantum
information processing. While most experimental demonstrations of
high-dimensional entanglement rely on fidelity-based witnesses, these are
powerless to detect entanglement within a large class of entangled quantum
states, the so-called unfaithful states. In this paper, we introduce a highly
flexible automated method to construct optimal tests for entanglement detection
given a bipartite target state of arbitrary dimension, faithful or unfaithful,
and a set of local measurement operators. By restricting the number or
complexity of the considered measurement settings, our method outputs the most
convenient protocol which can be implemented using a wide range of experimental
techniques such as photons, superconducting qudits, cold atoms or trapped ions.
With an experimental quantum optics setup that can prepare and measure
arbitrary high-dimensional mixed states, we implement some $3$-setting
protocols generated by our method. These protocols allow us to experimentally
certify 2- and 3-unfaithful entanglement in 4-dimensional photonic states, some
of which contain well above 50% of noise.
