Quantum transition probability ($P_{i\to f}(t)=|⟨f|U(t)|i⟩{|}^2$)

For a prepared normalized pure state $|i⟩$ and a final projective measurement onto the normalized state $|f⟩$, unitary time evolution gives the displayed probability, by the Born rule. This depends on the preparation, final measurement, and time-evolution operator; it does not assume a classical Markov chain of transitions.