  There are many ways to decompose the Hilbert space H of a composite quantum
system into tensor product subspaces. Different subsystem decompositions
generally imply different interaction Hamiltonians V, and therefore different
expectation values for subsystem observables. This means that thE uniqueness of
physical predictions is not guaranteed, despite the uniqueness of the total
Hamiltonian H and the total Hilbert space. Here we use Clausius' version of the
second law of thermodynamics (CSL) and standard identifications of
thermodynamic quantities to identify possible subsystem decompositions. It is
shown that agreement with the CSL is obtained, whenever the total Hamiltonian
and the subsystem-dependent interaction Hamiltonian commute (i.e. [H,V]=0). Not
imposing this constraint can result in the transfer of heat from a cooler to a
hotter subsystem, in conflict with thermodynamics. We also investigate the
status of the CSL with respect to non- standard definitions of thermodynamic
quantities and quantum subsystems.
