  A key question in the thermodynamics of open quantum systems is how to
partition thermodynamic quantities such as entropy, work, and internal energy
between the system and its environment. We show that the only partition under
which entropy is non-singular is based on a partition of Hilbert-space, which
assigns half the system-environment coupling to the system and half to the
environment. However, quantum work partitions non-trivially under Hilbert-space
partition, and we derive a Work Sum Rule that accounts for quantum work at a
distance. All state functions of the system are shown to be path independent
once this nonlocal quantum work is properly accounted for. The thermodynamics
of two classes of quasi-statically driven open quantum systems is analyzed:
systems with a finite environment in the grand canonical ensemble, and systems
with an unbounded environment. Our results are illustrated with applications to
a time-dependent two-level system and the driven resonant-level model.
