= Solution
Let $R=|\psi\rangle\langle\psi|$ and choose an <orthonormal basis> $\{|e_\mu\rangle\}$ of $\mathcal H_2$. The <reduced density matrix> on $S_1$ is the <partial trace>
$$
\boxed{\rho_1=\operatorname{Tr}_2R=\sum_\mu(I_1\otimes\langle e_\mu|)R(I_1\otimes|e_\mu\rangle).}
$$
Here $I_1$ is the identity on $\mathcal H_1$, and the bras and kets in the sum contract only the second tensor factor. Equivalently, if $|\psi\rangle=\sum_{a,\mu}c_{a\mu}|f_a\rangle\otimes|e_\mu\rangle$, then $(\rho_1)_{ab}=\sum_\mu c_{a\mu}c_{b\mu}^*$. The definition is independent of the chosen basis: it is characterized by $\operatorname{Tr}(M\rho_1)=\operatorname{Tr}[(M\otimes I_2)R]$ for every operator $M$ on $\mathcal H_1$.
The <generalized measurement postulate> takes $A_i$ to be operators on $\mathcal H_2$ satisfying $\sum_iA_i^\dagger A_i=I_2$. The <Born rule> and the conditional state update give
$$
\boxed{p_i=\langle\psi|I_1\otimes A_i^\dagger A_i|\psi\rangle,\qquad |\psi_i\rangle=\frac{(I_1\otimes A_i)|\psi\rangle}{\sqrt{p_i}}\quad(p_i>0).}
$$
If the outcome is not supplied to $S_1$, the appropriate post-measurement <density matrix> is $R'=\sum_i(I_1\otimes A_i)R(I_1\otimes A_i^\dagger)$. For any local operator $M$, cyclicity of the full <trace> gives
$$
\begin{aligned}
\operatorname{Tr}[(M\otimes I_2)R']
&=\sum_i\operatorname{Tr}[(M\otimes A_i^\dagger A_i)R]\\
&=\operatorname{Tr}[(M\otimes I_2)R].
\end{aligned}
$$
Since this holds for every $M$, $\boxed{\operatorname{Tr}_2R'=\rho_1}$. This is the <no-communication theorem>.
The invariance concerns the nonselective <quantum measurement>. A conditional <reduced density matrix> $\rho_{1|i}=\operatorname{Tr}_2|\psi_i\rangle\langle\psi_i|$ can change, although $\sum_i p_i\rho_{1|i}=\rho_1$. For example, measuring one half of a <Bell pair> in the <computational basis> changes the other half conditionally to $|0\rangle$ or $|1\rangle$; before learning the outcome, its <density matrix> remains $I/2$. Thus distant <quantum measurements> cannot transmit a controllable signal without classical communication. This <quantum no-signalling> is compatible with <special relativity>, even though some <entangled> states have correlations incompatible with a <local hidden-variable theory>.
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