Suppose I have two compartments of volume V separated by a barrier. Each one has an ideal gas, pure, of N molecules, and with equal pressure and temperature. However, I don't know whether both gases are the same one or different ones.
I remove the barrier, and wait until equilibrium is reached.
If the gases were different, theory predicts the change of entropy is $\Delta S=-2Nk(\ln [V/2V])$
If they were the same, theory predicts it is $\Delta S = 0$, since I didn't actually mix anything, and there is no change in state.
Can I somehow experimentally measure this $\Delta S$?
Not predict it from theory, but calculate it from real actual thermodynamic measurements, and without being able to identify the gases, to see whether I obtain $0$ or another number.
If I can't, then it seems to me it's an empty concept.