  Generalized probabilistic theories (GPT) provide a framework in which one can
formulate physical theories that includes classical and quantum theories, but
also many other alternative theories. In order to compare different GPTs, we
advocate an approach in which one views a state in a GPT as a resource, and
quantifies the cost of interconverting between different such resources. We
illustrate this approach on polygon theories (Janotta et al. New J. Phys 13,
063024, 2011) that interpolate (as the number n of edges of the polygon
increases) between a classical trit (when n=3) and a real quantum bit (when
n=infinity). Our main results are that simulating the transmission of a single
n-gon state requires more than one qubit, or more than log(log(n)) bits, and
that n-gon states with n odd cannot be simulated by n'-gon states with n' even
(for all n,n'). These results are obtained by showing that the classical
capacity of a single n-gon state with n even is 1 bit, whereas it is larger
than 1 bit when n is odd; by showing that transmitting a single n-gon state
with n even violates information causality; and by showing studying the
communication complexity cost of the nondeterministic not equal function using
n-gon states.
