I have recently asked somewhat related question and got very illuminating answer. After some thinking however I have realized that (at least) one more point is unclear to me:
How can we check whether given bipartite entangled mixed state has classical correlations between its subsystems?
This is what I know about the subject so far. Suppose we are given some mixed state: $$\rho = \sum_{i} p_i |\psi_i\rangle\langle\psi_i|$$ here $\{p_i \}$ defines probabilistic mixture of pure states $\{|\psi_i\rangle \} $. This state could be one of three:
- simply separable(product state) $ \rho = \rho_A \otimes \rho_B$. In this case there are no correlations(either classical or quantum) between subsystems.
- separable state $$\rho = \sum_k p_k \rho_A ^k \otimes \rho_B ^k \quad \sum_k p_k =1 \quad (1) $$ Firstly by definition this state has no quantum correlations between subsystems A and B. Secondly this state can be obtained from product state by LOCC. This means that given state has classical correlations between subsystems.
- Entangled state In this case given state can't be written in form (1). It will obviously have quantum correlations between subsystems. Yet what can we say about classical correlations between parts A and B? My guess would be to check whether this state can only be constructed out of product state by combination of both nonlocal operators $U_A \otimes U_B$ and LOCC. If LOCC is necessary then our state has classical correlations between subsystems. However this criterion(if correct at all) seems to be almost impossible to apply in real situation. Is there any other way to resolve the question?