  Decoherence of a center spin or qubit in a spin bath is essentially
determined by the many-body bath evolution. The bath dynamics can start either
from a pure state or, more generally, from a statistical ensemble. In the
preceding article [W. Yang and R. B. Liu, Phys. Rev. B \textbf{78}, 085315
(2008)], we have developed the cluster-correlation expansion (CCE) theory for
the so-called single-sample bath dynamics initiated from a factorizable pure
state. Here we present the ensemble CCE theory, which is based on similar ideas
of the single-sample CCE: The bath evolution is factorized into the product of
all possible cluster correlations, each of which accounts for the authentic
(non-factorizable) collective excitation of a group of bath spins, and for the
finite-time evolution in the qubit decoherence problem, convergent results can
be obtained by truncating the ensemble CCE by keeping cluster correlations up
to a certain size. A difference between the ensemble CCE and single-sample CCE
is that the mean-field treatment in the latter formalism of the diagonal part
of the spin-spin interaction in the bath is not possible in the former case.
The ensemble CCE can be applied to non-factorizable initial states. The
ensemble CCE is checked against the exact solution of an XY spin bath model.
For small spin baths, it is shown that single-sample dynamics is sensitive to
the sampling of the initial state from a thermal ensemble and hence very
different from the ensemble average.
