How to Partition a Quantum Observable

How to partition a non-local observable is a key question in the study of open quantum systems. Here we argue that any local description of a quantum observable should be inherited from the division of the underlying Hilbert space. This idea is formalized by the construction of a general procedure for the partitioning of quantum observables over orthogonal subspaces of a global Hilbert space. Moreover, it is shown that this method allows for the definition of density and current density operators for generic, non-local observables. The energy, particle and entropy current densities are described for a non-equilibrium quantum system. Finally, we show that this Hilbert space partitioning is the only partition of the von Neumann entropy which is consistent with the laws of thermodynamics.