Module Code: PHYS$120001$

$($c$)$ Suppose you have copies of a Quantum Box that performs a unitary rotation $R()$ of quantum states through an unknown angle $.$ We have $5$ of these boxes combined in series and fire $1000$ bullets through the upper slot of the first box. The bullets leave the last box and hit a wall at either location $U$ or location $L$ as shown in the figure.

cdots$$ubrace$(R()ubrace{)}_{R()}$

i$.$ What values of $$ would send all of the bullets into the lower location $L$ of the wall?

ii$.$ Suppose we set $$ to a value that sends all the bullets to $L$ as in part $($i$),$ but now a spy switches on a weak metal detector where the upper output slot of the $4$ th box meets the upper input slot of the $5$ th box. What difference will this make? Explain, without calculation, your reasoning.