How to determine the purity and nature of a composite system? So for the combined state of a pair of two-level atoms, A and B, with a density matrix
$$\rho =\frac{1}{2}\lvert g_A, g_B \rangle\langle g_A, g_B \rvert + \frac{1}{2}\lvert g_A, e_B \rangle\langle g_A, e_B \rvert \tag{1}$$
Where g and e denotes the ground and excited state, respectively. I have calculated the reduced matrix operator for system A by taking the partial trace with respect to B
$$ \rho_A = Tr_B(\rho)=\lvert g_A \rangle \langle g_A \rvert \tag{2}$$
Assuming I have calculated this correctly, I would like to ask the following:
(1) Is the purity of system A simply equal to 1 as it only contains one pair of a kat-bra?
(2) Regarding the state of the two atoms, how can I establish if the combined state is entangled or a product state from using the reduced density matrices of each system?
Thank you.
EDIT: Clarified my second question. Also, here is the reduced matrix operator I calculated for system B in case it is relevant to my second question
$$ \rho_B = Tr_A(\rho)=\frac{1}{2}(\lvert g_B \rangle \langle g_B \rvert +\lvert e_B \rangle \langle e_B \rvert) \tag{3}$$
 A: Question (1)
Yes, $A$ is in a pure state. By definition, a state $\rho$ is pure if and only if it is a projector, that is $\rho^2 = \rho$. This is equivalent to $\mathrm{Tr}(\rho^2) = 1$. From this, it is clear that $\rho_A$ is a pure state.
Question (2)
I suppose you want an explanation of the definitions behind the notion of entanglement. If $\rho$ is the state of a composite system made of two subsystems $A$ and $B$, we distinguish three types of correlation that may exist between the two subsystem, which correspond to three types of states $\rho$: factorized states (1), separable states (2) and, finally, entangled states (3). In the following explanation we will use $O_A$ and $O_B$ to designate generic observables of system $A$ and $B$ respectively.


*

*Factorized states are also called product states or simply separable states. We say that $\rho$ is factorized if it is the tensor product of two states: $$ \rho = \rho_A \otimes \rho_B . $$
In this case the expected value of the product observable $O_AO_B$ is simply $$ \left< O_A O_B \right>_\rho = \left< O_A \right>_{\rho_A} \left< O_B \right>_{\rho_B} . $$
That is, the two subsystems are statistically independent. You can easily check if a state $\rho$ is factorized: just compute $\rho_A = \mathrm{Tr}_B(\rho)$ and $\rho_B = \mathrm{Tr}_A(\rho)$, then calculate $\rho' = \rho_A \otimes \rho_B$. If $\rho' = \rho$, then $\rho$ is factorized. Note that the $\rho$ of your example is factorized. Therefore, as we said, the subsystems are statistically independent. In this case there is no correlation between $A$ and $B$, so there is obviously no entanglement.

*Separable states are also called classically correlated states. A state $\rho$ is said to be separable if it can be written as a convex combination of factorized states, that is $$ \rho = \sum_i p_i \rho_A^{(i)} \otimes \rho_B^{(i)} , $$ with $p_i>0$ and $\sum_i p_i = 1$. In this case the expected value of $O_AO_B$ is $$ \left< O_A O_B \right>_\rho = \sum_i p_i \left< O_A \right>_{\rho_A^{(i)}} \left< O_B \right>_{\rho_B^{(i)}} . $$
This time the two subsystems are correlated, but we can say that this correlation is "classical" in some sense: the statistics of the system can be interpreted as if we knew that


*

*with probability $p_1$, system $A$ is in state $\rho_A^{(1)}$ and at the same time system $B$ is in $\rho_B^{(1)}$;

*with probability $p_2$, system $A$ is in state $\rho_A^{(2)}$ and at the same time system $B$ is in $\rho_B^{(2)}$;

*with probability $p_3$, system $A$ is in state $\rho_A^{(3)}$ and at the same time system $B$ is in $\rho_B^{(3)}$;

*... (and so on).


So there is nothing inherently quantum mechanical in the correlations between $A$ and $B$. They can be understood with classical reasoning. Of course, we are only talking about the correlations existing between $A$ and $B$ as long as the state is $\rho$. The inner workings of $A$ and $B$, as well as the future evolution of the composite system are still quantum mechanical in nature.

*All the states that do not correspond to the last two definitions are said to be entangled. In this case, the correlations between $A$ and $B$ cannot be interpreted with "classical reasoning".
