Circuit that measures PVM

Question

How can I construct a circuit that measures PVM $\{|\psi\rangle\langle\psi|\},|\psi^\perp\rangle\langle\psi^\perp| \}$ where $|\psi\rangle=\cos\frac{\theta}{2}+e^{i\phi}\sin\frac{\theta}{2}|1\rangle$ by using only computational basis measurement and Pauli-axis rotation.

Answer 1

Recall that

$$ R_Y(\theta) = \begin{pmatrix} \cos\frac{\theta}{2} &-\sin\frac{\theta}{2} \\ \sin\frac{\theta}{2} &\cos\frac{\theta}{2} \end{pmatrix} \\ R_Z(\phi) = \begin{pmatrix} e^{-\frac{i\phi}{2}} & 0 \\ 0 & e^{\frac{i\phi}{2}} \end{pmatrix}. $$

Define

$$ U(\theta, \phi) = R_Z(\phi) R_Y(\theta) \equiv\begin{pmatrix} \cos\frac{\theta}{2} & -\sin\frac{\theta}{2} \\ e^{i\phi}\sin\frac{\theta}{2} & e^{i\phi}\cos\frac{\theta}{2} \end{pmatrix} $$

where $\equiv$ denotes equivalence up to unobservable global phase. We see that $U|0\rangle = |\psi\rangle$ and $U|1\rangle = |\psi^\perp\rangle$. Therefore

$$ |\psi\rangle\langle\psi| = U|0\rangle\langle 0|U^\dagger \\ |\psi^\perp\rangle\langle\psi^\perp| = U|1\rangle\langle 1|U^\dagger. $$

We conclude that to reproduce output statistics of the PVM $\{|\psi\rangle\langle\psi|\},|\psi^\perp\rangle\langle\psi^\perp| \}$ we should apply $U^\dagger$ followed by a measurement in the computational basis. If we also want to obtain the appropriate post-measurement state we then need to apply $U$.