  The framework of generalized probabilistic theories (GPTs) is a popular
approach for studying the physical foundations of quantum theory. The standard
framework assumes the no-restriction hypothesis, in which the state space of a
physical theory determines the set of measurements. However, this assumption is
not physically motivated. We generalize the framework to account for systems
that do not obey the no-restriction hypothesis. We then show how our framework
can be used to describe new classes of probabilistic theories, for example
those which include intrinsic noise. Relaxing the restriction hypothesis also
allows us to introduce a 'self-dualization' procedure, which yields a new class
of theories that share many features of quantum theory, such as obeying
Tsirelson's bound for the maximally entangled state. We then characterize joint
states, generalizing the maximal tensor product. We show how this new tensor
product can be used to describe the convex closure of the Spekkens toy theory,
and in doing so we obtain an analysis of why it is local in terms of the
geometry of its state space. We show that the unrestricted version of the
Spekkens toy theory is the theory known as 'boxworld' that allows maximal
nonlocal correlations.
