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L07E11.tex
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L07E11.tex
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\documentclass[solutions.tex]{subfiles}
\xtitle
\begin{document}
\maketitle
\begin{exercise} Calculate Alice's density matrix for
$\sigma_z$ for the "near-singlet" state.
\end{exercise}
The "near-singlet" state is characterized by the following state-vector:
\[
\ket{\Psi} = \sqrt{0.6}\ket{ud} - \sqrt{0.4}\ket{du}
\]
Alice's density matrix is defined by its components in Eq. $7.20$%
\footnote{p$205$, section $7.5$ \textit{Entanglement for Two Spins}}:
\[
\rho_{a'a} = \sum_b\psi^*(a,b)\psi(a',b)
\]
Where $\psi(a,b)$ is the wave-function of the composite system, that we
can extract from $\ket{\Psi}$:
\[
\ket{\Psi} = \psi(u,u)\ket{uu}
+ \psi(u,d)\ket{ud}
+ \psi(d,u)\ket{du}
+ \psi(d,d)\ket{dd}
\Rightarrow \begin{cases}
\psi(u,u) = \psi(d,d) &= 0 \\
\psi(u,d) &= \sqrt{0.6} \\
\psi(d,u) &= -\sqrt{0.4} \\
\end{cases}
\]
Hence:
\[
\rho = \begin{pmatrix}
\rho_{uu} & \rho_{ud} \\
\rho_{du} & \rho_{dd} \\
\end{pmatrix} = \begin{pmatrix}
\psi^*(u,u)\psi(u,u)+\psi^*(u,d)\psi(u,d)
& \psi^*(d,u)\psi(u,u)+\psi^*(d,d)\psi(u,d) \\
\psi^*(u,u)\psi(d,u) + \psi^*(u,d)\psi(d,d)
& \psi^*(d,u)\psi(d,u) + \psi^*(d,d)\psi(d,d)
\end{pmatrix} = \boxed{\begin{pmatrix}
0.6 & 0 \\
0 & 0.4 \\
\end{pmatrix}} \qed
\]
\begin{remark} I'm not sure what the authors expect regarding
$\sigma_z$; we're asked to verify all numerical values in
the next exercise, which likely should cover pretty much every
intepretation (we'll even have to compute Alice's density matrix
again, so as to check $\rho^2$/$\Tr(\rho^2)$).
\end{remark}
\end{document}