# Density matrix of spatially (i.e., at the same time instant) vs. causally ( i.e., one evolves into the other) correlated quantum systems

In the classical case, if Y is the output of a classical channel whose input is X, it makes sense to speak of a joint distribution $$P_{XY}$$. In the quantum case, if a state $$\rho_A$$ is input to a channel $$\mathcal{E}_{B|A}$$ producing $$\rho_B$$, speaking of a quantum state $$\rho_{AB}$$ is not consistent. However, if A and B are spatially correlated systems, and it makes sense in that case to speak of $$\rho_{AB}$$. Why is that?

• You might find this paper interesting – Rammus Jun 23 at 17:32
• Yes, that's what I am looking for, thank you. – Dina Abdelhadi Jun 24 at 6:42