I hope that the following experiment will help me understand the topic better. Let's say my friend is sending me photons via two channels and he is doing it in one of the following two ways:
he is sending one photon at the time in one of the two channels. Sometimes through first, sometimes through second channel(i dont know which one unless i measure them).
he is using a beam splitter and sending photons in superposition of through two channels, but he also adds some unknown phase to one of the channels each time he send a photon.
So I am receiving photons and I can do whatever I want with them. How could I determine whether he is doing option 1. or option 2. Since he is adding some different phase each time I guess that by recombine two paths I could not conclude much
EDIT: After reading the answers by Ronak and Julien I figured I have one more question that is puzzling me. In case there is no experiment that can distinguish two options, can I say that the states describing them are the same? Are they also the same for my friend how knows how much phase is changed each time?