Distinguishable Terms in a State

[Swamp Thing]

Let's say we have a proposition (or state, if we prefer) $P\equiv \psi \equiv AB^\perp + e^{j \theta} A^\perp B$ where, e.g., $A^{\perp}$ indicates some ket that is orthogonal to $A$.

We also have an operator $\hat{O}$ .

Without reference to a physical context, is there a test to say whether there will be interference in the probability of observing a given eigenvalue of $\hat{O}$ as we vary the phase $\theta$ ? In other words, how would a shut-up-and-calculate robot know whether it should sum-then-square or it should square-then-sum?

 

[MichPod]

We need to get a projection operator $\hat\Pi$ corresponding to a given eigenvalue of $\hat O$. Then the probability is calculated as $$ <\psi|\hat\Pi|\psi>$$ If it changes when θ changes, then we have interference.