Quantum Miniatures: quantum-like theories with limited sets of observables

A framework is introduced for constructing models – called quantum miniatures
– which structurally resemble standard quantum systems. To construct quantum
miniatures, we consider an ordinary quantum system but restrict the set of physical
observables, in order to examine the implications of the commonly made assumption
that all self-adjoint operators represent observables.
We investigate various consequences of placing a restriction on observables, which
allows for a natural extension of the state space of a miniature beyond the usual
quantum states. Models are developed which maximally violate uncertainty relations
both for spin variables and for position and momentum, allowing for simultaneous
sharp prediction of non-commuting observables. Another model exhibits postquantum
non-locality, including PR box correlations.
The quantum miniature framework includes standard quantum theory as a special
case, and allows us to define consistent foil theories to quantum theory, including
providing an explicit physical setting for a well-known family of GPTs, the polygon
models of Janotta et al.