Question

A system of two spin- \( 1 / 2 \mathrm{~s} A \) and \( B \) is in the state \[ \left|\psi_{A+B}\right\rangle=\alpha_{++}|++\rangle+\alpha_{+-}|+-\rangle+\alpha_{-+}|-+\rangle+\alpha_{--}|--\rangle . \] Show that \[ \begin{aligned} \hat{\rho}^{(\mathrm{A})}= & \left(\left|\alpha_{++}\right|^{2}+\left|\alpha_{+-}\right|^{2}\right)|+\rangle\langle+|+\left(\left|\alpha_{--}\right|^{2}+\left|\alpha_{-+}\right|^{2}\right)|-\rangle\langle-| \\ & +\left(\alpha_{++} \alpha_{-+}^{}+\alpha_{+-} \alpha_{--}^{}\right)|+\rangle\langle-|+\left(\alpha_{--} \alpha_{+-}^{}+\alpha_{-+} \alpha_{++}^{}\right)|-\rangle\langle+| . \end{aligned} \] and \[ \begin{aligned} \hat{\rho}^{(\mathrm{B})}= & \left(\left|\alpha_{++}\right|^{2}+\left|\alpha_{-+}\right|^{2}\right)|+\rangle\langle+|+\left(\left|\alpha_{--}\right|^{2}+\left|\alpha_{+-}\right|^{2}\right)|-\rangle\langle-| \\ & +\left(\alpha_{++} \alpha_{+-}^{}+\alpha_{-+} \alpha_{--}^{}\right)|+\rangle\langle-|+\left(\alpha_{--} \alpha_{-+}^{}+\alpha_{+-} \alpha_{++}^{}\right)|-\rangle\langle+| . \end{aligned} \]

          A system of two spin- \( 1 / 2 \mathrm{~s} A \) and \( B \) is in the state
\[
\left|\psi_{A+B}\right\rangle=\alpha_{++}|++\rangle+\alpha_{+-}|+-\rangle+\alpha_{-+}|-+\rangle+\alpha_{--}|--\rangle .
\]

Show that
\[
\begin{aligned}
\hat{\rho}^{(\mathrm{A})}= & \left(\left|\alpha_{++}\right|^{2}+\left|\alpha_{+-}\right|^{2}\right)|+\rangle\langle+|+\left(\left|\alpha_{--}\right|^{2}+\left|\alpha_{-+}\right|^{2}\right)|-\rangle\langle-| \\
& +\left(\alpha_{++} \alpha_{-+}^{*}+\alpha_{+-} \alpha_{--}^{*}\right)|+\rangle\langle-|+\left(\alpha_{--} \alpha_{+-}^{*}+\alpha_{-+} \alpha_{++}^{*}\right)|-\rangle\langle+| .
\end{aligned}
\]
and
\[
\begin{aligned}
\hat{\rho}^{(\mathrm{B})}= & \left(\left|\alpha_{++}\right|^{2}+\left|\alpha_{-+}\right|^{2}\right)|+\rangle\langle+|+\left(\left|\alpha_{--}\right|^{2}+\left|\alpha_{+-}\right|^{2}\right)|-\rangle\langle-| \\
& +\left(\alpha_{++} \alpha_{+-}^{*}+\alpha_{-+} \alpha_{--}^{*}\right)|+\rangle\langle-|+\left(\alpha_{--} \alpha_{-+}^{*}+\alpha_{+-} \alpha_{++}^{*}\right)|-\rangle\langle+| .
\end{aligned}
\]

A system of two spin- 1 / 2 s A and B is in the state

|ψA+B⟩=α++|++⟩+α+-|+-⟩+α-+|-+⟩+α–|–⟩ .

Show that

ρ̂^(A)= (|α++|^2+|α+-|^2)|+⟩⟨+|+(|α–|^2+|α-+|^2)|-⟩⟨-|
+(α++α-+^*+α+-α–^*)|+⟩⟨-|+(α–α+-^*+α-+α++^*)|-⟩⟨+| .

and

ρ̂^(B)= (|α++|^2+|α-+|^2)|+⟩⟨+|+(|α–|^2+|α+-|^2)|-⟩⟨-|
+(α++α+-^*+α-+α–^*)|+⟩⟨-|+(α–α-+^*+α+-α++^*)|-⟩⟨+| .

Added by Cristian H.

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