Generalized probabilistic theory
A generalized probabilistic theory (GPT) is a general framework to describe the operational features of arbitrary physical theories. A GPT must specify what kind of physical systems one can find in the lab, as well as rules to compute the outcome statistics of any experiment involving labeled preparations, transformations and measurements. The framework of GPTs has been used to define hypothetical non-quantum physical theories which nonetheless possess quantum theory's most remarkable features, such as entanglement or teleportation. Notably, a small set of physically motivated axioms is enough to single out the GPT representation of quantum  theory. Frameworks similar to that of GPTs have been used since at least the 1960s, but the term "generalized probabilistic theory" itself was coined by Jonathan Barrett in 2007, based on the framework introduced by Lucien Hardy. Note that some authors also use the term "operational probabilistic theory" to denote a particular variant of GPTs. Definition. A GPT is specified by a number of mathematical structures, which are: It can be argued that if one can prepare a state formula_1 and a different state formula_2, then one can also toss a (possibly biased) coin which lands on one side with  probability formula_3 and on the other with probability formula_4 and prepare either formula_1 or formula_2, depending on the side the coin lands on. The resulting state is a statistical mixture of the states formula_1 and formula_2 and in GPTs such statistical mixtures are described by convex combinations, in this case formula_9. For this reason all state spaces are assumed to be convex sets. Following a similar reasoning, one can argue that also the set of measurement outcomes and set of physical operations must be convex. Additionally it is always assumed that measurement outcomes and physical operations are affine maps, i.e.  that if formula_10 is a physical transformation, then we must have formula_11and similarly for measurement outcomes. This follows from the argument that we should obtain the same outcome if we first prepare a statistical mixture and then apply the physical operation, or if we prepare a statistical mixture of the outcomes of the physical operations. Note that physical operations are a subset of all affine maps which transform states into states as we must require that a physical operation yields a valid state even when it is applied to a part of a system (the notion of "part: is subtle : it is specified by explaining how different system types compose and how the global parameters of the composite system are affected by local operations). For practical reasons it is often assumed that a general GPT is embedded in a finite-dimensional vector space, although infinite-dimensional formulations exist. Classical, quantum, and beyond. Classical theory is a GPT where states correspond to probability distributions and both measurements and physical operations are stochastic maps. One can see that in this case all state spaces are simplexes. Quantum theory is a GPT where system types are described by a natural number formula_12 which  corresponds to the Hilbert space dimension. States of the systems of Hilbert space dimension formula_12 are described by the normalized positive semidefinite matrices, i.e. by the density matrices. Measurements are identified with Positive Operator valued Measures (POVMs), and the physical operations are completely positive maps. Systems compose via the tensor product of the underlying Hilbert spaces. The framework of GPTs has provided examples of consistent physical theories which cannot be embedded in quantum theory and indeed exhibit very non-quantum features. One of the first ones was Box-world, the theory with maximal non-local correlations. Other examples are  theories with third-order interference and the family of GPTs known as generalized bits. Many features that were considered purely quantum are actually present in all non-classical GPTs. These include the impossibility of universal broadcasting, i.e., the no-cloning theorem; the existence of incompatible measurements; and the existence of entangled states or entangled measurements.