  In classical finite-range spin systems, especially those with disorder such
as spin glasses, a low-temperature Gibbs state may be a mixture of a number of
pure or ordered states; the complexity of the Gibbs state has been defined in
the past roughly as the logarithm of this number, assuming the question is
meaningful in a finite system. As non-trivial mixtures of pure states do not
occur in finite size, in a recent paper [Phys. Rev. E 101, 042114 (2020)]
H\"oller and the author introduced a definition of the complexity of an
infinite-size Gibbs state as the mutual information between the pure state and
the spin configuration in a finite region, and applied this also within a
metastate construction. (A metastate is a probability distribution on Gibbs
states.) They found an upper bound on the complexity for models of Ising spins
in which each spin interacts with only a finite number of others, in terms of
the surface area of the region, for all $T\geq 0$. In the present paper, the
complexity of a metastate is defined likewise in terms of the mutual
information between the Gibbs state and the spin configuration. Upper bounds
are found for each of these complexities for general finite-range (i.e. short-
or long-range, in a sense we define) mixed $p$-spin interactions of discrete or
continuous spins (such as $m$-vector models), but only for $T>0$. For
short-range models, the bound reduces to the surface area. For long-range
interactions, the definition of a Gibbs state has to be modified, and for these
models we also prove that the states obtained within the metastate
constructions are Gibbs states under the modified definition. All results are
valid for a large class of disorder distributions.
