Understanding the density matrix for pure states vs. mixed states
So, there's a question in my textbook that explains the following scenario:
A physicist runs two experiments A and B to prepare quantum systems in a variety of initial states. In experiment A he uses a probabilistic machine that can prepare a single quantum system in one of n possible pure states ${\lvert \psi_1 \rangle, \lvert \psi_2 \rangle, ..., \lvert \psi_n \rangle }$ with corresponding probabilities ${\lvert p_1 \rangle, \lvert p_2 \rangle, ..., \lvert p_n \rangle }$. In experiment B, instead, he generates m non-interacting quantum systems, each of them prepared in its corresponding lower energy state ${\lvert \phi_1 \rangle, \lvert \phi_2 \rangle, ..., \lvert \phi_m \rangle }$. Let $\rho_A$ and $\rho_B$ be the density matrix operators for the quantum states prepared in experiments A and B, respectively.
And asks the following:
1: Write down an expression for both $\rho_A$ and $\rho_B$ and derive an expression for the expected value of $\hat{O_A}$ and $\hat{O_B}$ (the Hermitian operators describing an observable of the system A and kth system of system B, respectively).
2: Let $P_j$ and $P_k$ be the projectors associated to the states $\lvert \psi_j \rangle$ and $\lvert \psi_k \rangle$ produced by A. 680
Discuss whether the product $P_jP_k$ vanishes.
For question 1, I wrote the following for system A:
$$\rho_A = \lvert \psi_j\rangle\langle \psi_j\rvert \tag{1}$$
$$\hat{O_A} = Tr(\hat{O_A} \rho_A) = Tr(\hat{O_A}\lvert\psi_j\rangle\langle \psi_j\rvert) \tag{2}$$
My justification for equation 1 is that system A produces pure states only and that only "a single" state is generated, so therefore the density matrix only contains one term (here j represents the jth state generated). However, the question mentions their probabilities which I haven't included in my answer because only one state is generated. Should $p_j$ be the coefficient in equation (1) as opposed to just the number 1?
For system B, I have the following:
$$\rho_B = \frac{1}{m}\sum_i\lvert \phi_i\rangle\langle \phi_i\rvert \tag{3}$$
My justification for this is that experiment B describes a mixed state, which produces a total of m states and therefore each state has a probability of $\frac{1}{m}$ . I am unsure, however, about how to write the expected value for experiment B.
For the second question, I can express the product as
$$ P_jP_k = \lvert \psi_j\rangle\langle \psi_j\rvert\lvert \psi_k\rangle\langle \psi_k\rvert \tag{4}$$
But as I'm given no information regarding the orthogonality of the states for experiment A, how am I supposed to deduce whether this equation vanishes or not?
If anyone could provide any hints or insight into my answers, it would be greatly appreciated. I am not fully confident that I correctly understand the differences between pure and mixed states. 680