Entropy of mixing exists only for classical and quantum-like theories among the regular polygon theories

The thermodynamical entropy of a system which consists of different kinds of ideal gases is known to be defined successfully in the case when the differences are described by classical or quantum theory. Since these theories are special examples in the framework of generalized probabilistic theories (GPTs), it is natural to generalize the notion of thermodynamical entropy to systems where the internal degrees of particles are described by other possible theories. In this paper, we consider thermodynamical entropy of mixing in a specific series of theories of GPTs called the regular polygon theories, which can be regarded from a geometrical perspective as intermediate theories between a classical trit and a quantum bit with real coefficients. We prove that the operationally natural thermodynamical entropy of mixing does not exist in those inbetween theories, that is, the existence of the natural entropy results in classical and quantum-like theories among the regular polygon theories. (paper)