On the distinguishability of downconverted modes with orbital angular momentum

Assuming two quantum states of spontaneous parametric downconversion carrying orbital angular momentum, one may ask the question what is the minimum probability of error in identifying between two of these biphoton states by an arbitrary physical measurement over the biphoton state generated. While correctly chosen geometries may lead to perfect distinguishability of modes, it is worth noticing that experimental subtleties may lead to a poor mode distinguishability. We discuss the case where a restricted range instead of the needed range of wave vectors is collected by the experimental setup. These considerations may be useful for some applications, e.g., cryptography.