1. The problem statement, all variables and given/known data Consider a qubit whihc undergoes a sequence of three reversible evolutions of 3 unitary matrices A, B, and C (in that order). Suppose that no matter what the initial state |v> of the qubit is before the three evolutions, it always comes back to the sam state |v> after the three evolutions. Show that we must have C=(BA)† 2. Relevant equations † = hermitian conjugate 3. The attempt at a solution The diagram of the reversible evolution allows us to see that the process |v> --> A --> B --> C = |v> results in the equation: C(BA)|v> = |v> From here we see that: C(BA) = I (where I is the identity matrix) We multiple both sides by (BA)† C(BA)(BA)†=I(BA)† By definition of unitary we see C=(BA)† This was quite easy, we see it only took 3-4 steps. Have I successfully completed this proof? Recall, I needed to show that this works for all possible |v>. |v> seems irrelevant, however, since it could be 'cancelled' out in the second step. Success? Or failure?