HW1 - Problem 3

Problem 3: More bombs

Suppose we repeat the Elitzur-Vaidman bomb experiment but adjust our circuit as follows:

The final measurement is in the standard basis. Recall that

\(R_\theta := \begin{bmatrix}\cos \theta & - \sin \theta \\ \sin \theta & \cos \theta\end{bmatrix}.\)

To recap, the action of the “bomb” gate is the following: - If the bomb is faulty, it does nothing and acts as an identity gate. - If the bomb is not faulty, and the input state is \(\alpha \ket{0} + \beta\ket{1}\), with probability \(|\beta|^2\) the bomb explodes and the experiment ends. With probability \(|\alpha|^2\), nothing happens and the state is reset to \(\ket{0}\).

Just like in class, there are three possible outcomes. - The bomb explodes - We measure \(\ket{0}\) - We measure \(\ket{1}\)

  1. What are the possible outcomes and probabilities of those outcomes if the bomb is faulty?
  2. What are the possible outcomes and probabilities of those outcomes if the bomb is not faulty?
  3. How does this version of the experiment compare to what we did in class? Which is more effective at detecting a bomb that is not faulty without having it explode? Which has a higher probability that the bomb explodes?