Calculating the reduced density matrix of two separate experiments

Question

Consider setting up two experiments $alpha $ and $beta$.
In experimental setup $alpha$ we can prepare $n$ possible pure states ${| psi_1 rangle, | psi_2 rangle, ... | psi_n rangle}$ with associated probabilities ${p_1,p_2,...p_n}$.
In experimental setup $beta$ we prepare $m$ non-interacting systems. Each system  prepared in its corresponding lower energy states ${| phi_1 rangle, | phi_2 rangle, ... | phi_n rangle}$.
My question is how would one deduce the density matrix operators for the quantum states prepared in each experiment.
Is it simply a matter of noting that one can denote the density operator as  a sum of a set of projectors  as follows: $rho = sum_j p_j |jrangle langle j|.$
So then I may kind of assume that the density matrix operator for states prepared in $alpha $ will be $rho_{alpha} = sum_j p_i |psi_irangle langle psi_i|.$
But I am unsure if this is correct I am also unsure how to construct the equivalent for experiment $beta$.

pglpm · Accepted Answer

With some more thought you arrive at the answer for the second part of your question, since you have the correct understanding of the first part.
Setup $alpha$: we know one system has been prepared in one of the $n$ possible pure states ${lvertpsi_1rangle,dotsc,lvertpsi_nrangle}$. We don't know which, but we assign probabilities $p_1,dotsc,p_n$ to the $n$ possibilities. Then the density matrix representing our knowledge of this setup is, as you correctly wrote, $sum_i p_i lvertpsi_iranglelanglepsi_irvert$.
Setup $beta$: we know that $m$ systems have been prepared in $m$ pure states (possibly identical for some of them), each from among the set ${lvertphi_1rangle,dotsc,lvertphi_nrangle}$. We also know the preparation state of each system, say $lvertphi_{i_k}rangle$ for the $k$th system.
This means that their joint system has been prepared in the pure state $lvert phi_{i_1},dotsc,phi_{i_m}rangle$, which is equivalent to the density matrix $lvert phi_{i_1},dotsc,phi_{i_m}ranglelangle phi_{i_1},dotsc,phi_{i_m}rvert equiv lvert phi_{i_1}ranglelangle phi_{i_1}rvert otimes dotsb otimes lvert phi_{i_m}ranglelangle phi_{i_m}rvert$. (We're assuming that there are no bosonic or fermionic symmetrization complications.)
Saying that they're non-interacting is unimportant for the specification of the state (kinematics): that qualification refers to the dynamics of the joint system, meaning that it has the form $U_1 otimes dotsb otimes U_m$, where $U_k$ is the evolution operator acting on the $k$th system.
Just to add a reference, my favourite book about these topics: Bengtsson, Życzkowski: Geometry of Quantum States: An Introduction to Quantum Entanglement (2nd ed., Cambridge 2017).

Calculating the reduced density matrix of two separate experiments

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