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Alex, thanks for the recommendations. I'm actually going back and forth between the first two you suggested. Do you think one is better than the other?
So, would it be correct to say that even though $a_{1}|A\rangle$ is the same state as $a_{2}|A\rangle$ and $b_{1}|B\rangle$ is the same state as $b_{2}|B\rangle$, $|R_{1}\rangle$ and $|R_{2}\rangle$ represent two different ways of superposing these states, so that the probabilities of being in the state represented by $|A\rangle$ and $|B\rangle$ are different for each?
