Exercise 1
The inner product of \(\mathbf U \ket{A}\) and \(\mathbf U \ket{B}\) is \[ \braket{A | \mathbf U^\dagger \mathbf U | B } = \braket { A | \mathbf I | B } = \braket{A|B} \]
since \(\mathbf U\) is unitary.
The inner product of \(\mathbf U \ket{A}\) and \(\mathbf U \ket{B}\) is \[ \braket{A | \mathbf U^\dagger \mathbf U | B } = \braket { A | \mathbf I | B } = \braket{A|B} \]
since \(\mathbf U\) is unitary.