  The signaling dimension of a given physical system quantifies the minimum
dimension of a classical system required to reproduce all input/output
correlations of the given system. Thus, unlike other dimension measures - such
as the dimension of the linear space or the maximum number of (jointly or
pairwise) perfectly discriminable states - which examine the correlation space
only along a single direction, the signaling dimension does not depend on the
arbitrary choice of a specific operational task. In this sense, the signaling
dimension summarizes the structure of the entire set of input/output
correlations consistent with a given system in a single scalar quantity. For
quantum theory, it was recently proved by Frenkel and Weiner in a seminal
result that the signaling dimension coincides with the Hilbert space dimension.
Here, we derive analytical and algorithmic techniques to compute the signaling
dimension for any given system of any given generalized probabilistic theory.
We prove that it suffices to consider extremal measurements with ray-extremal
effects, and we bound the number of elements of any such measurement in terms
of the linear dimension. For systems with a finite number of extremal effects,
we recast the problem of characterizing the extremal measurements with
ray-extremal effects as the problem of deriving the vertex description of a
polytope given its face description, which can be conveniently solved by
standard techniques. For each such measurement, we recast the computation of
the signaling dimension as a linear program, and we propose a combinatorial
branch and bound algorithm to reduce its size. We apply our results to derive
the extremal measurements with ray-extremal effects of a composition of two
square bits (or squits) and prove that their signaling dimension is five, even
though each squit has a signaling dimension equal to two.
