What is this question about?
Consinder a system $\mathcal{AB}$, consisting of two subsystems $\mathcal{A}$ and $\mathcal{B}$ (with hilbert-spaces $\mathcal{H_A}$ and $\mathcal{H_B}$). I won't be introducing all formalisms of quantum mechanics here - I assume it is clear what is used in the following.
I am now trying to proof the following (=derive the reduced density matrix of the system $\mathcal{A}$):
If an observer only has only access to $\mathcal{A}$, there is a simpler mathematical object, that contains all information the observer can learn, than the density matrix $\rho$, namely the reduced density matrix.
Schlosshauer (ISBN: 978-3540357735) derives it on page 46 in the following way:
Following the above request, one must be able to calculate the expectation value of an Observable $O_{\mathcal{A}}\otimes I$ from this 'reduced density matrix'. We calculate with $\{\psi_k\}$ and $\{\phi_l\}$ orthonormal bases of the hilbert spaces introduced above:
$$\langle O\rangle=\text{Tr}(\rho O)=\\=\sum_{kl}\langle\phi_l|\langle\psi_k|\rho(O_{\mathcal{A}}\otimes I)|\psi_k\rangle|\phi_k\rangle=\\= \sum_k\langle\psi_k|\left(\sum_k\langle\phi_l|\rho|\phi_k\rangle\right)O_{\mathcal{A}}|\psi_k\rangle=\\= \sum_k\langle\psi_k|(\text{Tr}_{\mathcal{B}}\rho)O_{\mathcal{A}}|\psi_k\rangle=\\=\text{Tr}_{\mathcal{A}}(\rho_{\mathcal{A}}O_{\mathcal{A}})$$
with $\rho_{\mathcal{A}}$ the (as in the above calculation) defined reduced density matrix of the system $\mathcal{A}$.
What concerns do I have?
- Wrong definition of the reduced density matrix?: How can one calculate $\rho|\psi_k\rangle$? The dimensions of those elements (if seen as matrix and vector) don't match - in linear algebra this operation is not defined / would be forbidden!
- If the trace is defined by $\text{Tr}(\gamma)=\langle\alpha_i|\gamma|\alpha_i\rangle$ and $\langle\phi_l|\langle\psi_k|$ above means $\langle\phi_l|\otimes\langle\psi_k|$ then the definition of the trace is used in the wrong way, because $(|\phi_l\rangle\otimes|\psi_k\rangle)^{\dagger}=\langle\phi_l|\otimes\langle\psi_k|$ - so the second line above should rather read $$ \sum_{kl}\langle\psi_k|\langle\phi_l|\rho(O_{\mathcal{A}}\otimes I)|\psi_k\rangle|\phi_k\rangle.$$
What I'd like to know
So ultimately I fail to derive the reduced density matrix. I would gladly appreciate any hint on books that derive it properly.
I would like to ask anyone to explain to me, if my concerns are correct or if I am making a mistake there. If somebody is able to clear up Schlosshauers derivation for me, I'd also appreciate if you gave some references to the used mathematics. Thank you!