  A phase coexistence state cannot be specified uniquely by any intensive
parameters, such as the temperature and the magnetic field, because they take
the same values over all coexisting phases. It can be specified uniquely only
by an appropriate set of additive observables. Hence, to analyze phase
coexistence states the statistical ensembles that are specified by additive
observables have been employed, such as the microcanonical and restricted
ensembles. However, such ensembles are ill-defined or ill-behaved when some of
the additive observables do not commute with each other. Here, we solve this
fundamental problem by extending a generalized ensemble in such a way that it
is applicable to phase coexistence states which are specified by noncommutative
additive observables. We prove that this ensemble correctly gives the density
matrix corresponding to phase coexistence states of general quantum systems as
well as the thermodynamic functions. Furthermore, these ensembles are
convenient for practical calculations because of good analytic properties and
useful formulas by which temperature and other intensive parameters are
directly obtained from the expectation values of the additive observables. As a
demonstration, we apply our formulation to a two-dimensional system whose phase
coexistence states are specified by an additive observable (order parameter)
that does not commute with the Hamiltonian.
