Does this derivation of the reduced density matrix really prove it containing "all measurement statistics"?

Question

This question is a follow up to this question on the derivation of the reduced density matrix.

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What this question is about:

According to Schlosshauer (ISBN: 978-3540357735), the reduced density matrix is a mathematical object that contains all information that an observer could learn about a subsystem $\mathcal{A}$ of a system $\mathcal{AB}$ (so it contains all measurement-statistics). In his derivation of it, he proves that it allows the calculation of the expected value of an observable $O_{\mathcal{A}}\otimes I$ acting on $\mathcal{AB}$ - Indeed, he is looking for an object to do just that in order to find the reduced density matrix which (repeating myself) "contains all information that an observer could learn about a subsystem $\mathcal{A}$ of a system $\mathcal{AB}$ (so it contains all measurement-statistics)."

The question:

Is proving

that the reduced density matrix allows the calculation of above mentioned expectation value

enough to state that it contains all information an observer could learn from $\mathcal{A}$ ("all measurement statistics"? What I mean is that, while it is now proven that the expected value (above) can be calculated with the reduced density matrix, it is (in my opinion) not clear that one could calculate the state of $\mathcal{A}$ after the measurement - for either of the measurement results, which are defined by $O_{\mathcal{A}}\otimes I$.

Answer 1

The reduced density matrix allows to compute all expectation values for the subsystem alone, in other words: For all hermitian operators $O_A$ on $H_A$ we have that

$$ \mathrm{Tr}\, \rho \,O_A \otimes \mathbb I_B= \mathrm{Tr}^{(A)}\rho_A O_A \tag{1} \quad .$$

This, in particular, includes the calculation of probabilities of measurement results of local observables: Write the observable $O_A$ in its spectral representation:

$$O_A=\sum\limits_j o_j\, P_j \tag{2} \quad .$$

The probability to measure $o_j$ is then given by $$\mathrm{Tr}\,\rho\, P_j \otimes \mathbb I_B = \mathrm{Tr}^{(A)}\rho_A P_j \tag{3} \quad . $$ To emphasize: The left-hand side is a postulate, the right-hand side a trivial consequence of $(1)$ (which itself is a consequence of the definition of $\rho_A$). The state of the bipartite system after the measurement is $$ \rho^\prime \propto P_j \otimes \mathbb I_B \, \rho\,P_j \otimes \mathbb I_B \tag{4}$$ with the associated reduced density matrix

$$\rho_A^\prime= \mathrm{Tr}_B \,\rho^\prime \propto P_j\, \rho_A\, P_j \quad . \tag{5} $$ Again: Equation $(4)$ is a postulate, equation $(5)$ is a consequence of it. Indeed, the above follows from the observation that for $|\psi\rangle \in H_B$ it holds that

$$\left(P_j \otimes \mathbb I_B\right) \left(\mathbb I_A\otimes |\psi\rangle\right) =P_j \otimes |\psi\rangle = \left(\mathbb I_A \otimes |\psi\rangle\right)\, P_j \tag{6}$$

and $$ \left(\mathbb I_A\otimes \langle \psi| \right) \left(P_j \otimes \mathbb I_B\right) = P_j \otimes \langle \psi| = P_j\, \left(\mathbb I_A\otimes \langle \psi|\right) \quad . \tag{7} $$

Hence, the knowledge of $\rho_A$ suffices to compute all observable properties of the subsystem corresponding to $H_A$.

Source and further reading: J. Audretsch. Entangled Systems: new directions in quantum physics. Wiley, especially chapter 7. A pdf of the relevant chapter can be found here.