Ex4 20

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 4.20}$

Let $O_{\theta_1}$ be the single-qubit observable with $+1$-eigenvector
$\ket{v_1} = \cos\theta_1\ket{0} + \sin\theta_1\ket{1}$
and $-1$-eigenvector
$\ket{v_1^\perp} = -\sin_1\theta\ket{0} + \cos\theta_1\ket{1}.$
Similarly let $O_{\theta_2}$ be the single-qubit observable
with $+1$-eigenvector
$\ket{v_2} = \cos\theta_2\ket{0} + \sin\theta_2\ket{1}$
and $-1$-eigenvector
$\ket{v_2^\perp} = -\sin\theta_2\ket{0} + \cos\theta_2\ket{1}.$
Let $O$ be the two-qubit observable $O_{\theta_1}\otimes O_{\theta_2}$. We consider various measurements on the EPR state $\ket{\psi} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$. We are interested in the probability that the measurements $O_{\theta_1}\otimes I$ and $I\otimes O_{\theta_2}$, if they were performed on the state $\ket\psi$, would {\it agree} on the two qubits in that either both qubits are measured in the $1$-eigenspace or both are measured in $-1$-eigenspace of their respective single-qubit observables.
As in Example \ref{bit-equality}, we are not interested in the specific outcome of the two measurements, just whether or not they would agree. The observable $O = O_{\theta_1}\otimes O_{\theta_2}$ gives exactly this information.

a) Find the probability that the measurements $O_{\theta_1}\otimes I$ and $I \otimes O_{\theta_2}$, when performed on $\ket\psi$, would agree in the sense of both resulting in a $+1$ eigenvector or both resulting in a $-1$ eigenvector. (Hint: Use the trigonometric identities $\cos(\theta_1 - \theta_2) = \cos(\theta_1)\cos(\theta_2) + \sin(\theta_1)\sin(\theta_2)$ and $\sin(\theta_1 - \theta_2) = \sin(\theta_1)\cos(\theta_2) - \cos(\theta_1)\sin(\theta_2)$ to obtain a simple form for your answer.)

b) For what values of $\theta_1$ and $\theta_2$ do the results always agree?

c) For what values of $\theta_1$ and $\theta_2$ do the results never agree?

d) For what values of $\theta_1$ and $\theta_2$ do the results agree half the time?

e) Show that whenever $\theta_1 \ne \theta_2$ and $\theta_1$ and $\theta_2$ are chosen from $\{-60^\circ, 0^\circ, 60^\circ \}$, then the results agree $1/4$ of the time and disagree $3/4$ of the time.

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