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?

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

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