The Spekkens toy model is a conceptually simple toy hidden-variable theory introduced by Robert Spekkens in 2004, to argue in favour of the epistemic view of quantum mechanics.
Within the bounds of this model, many phenomena typically associated with strictly quantum-mechanical effects are present.
For nearly a century, physicists and philosophers have been attempting to explain the physical meaning of quantum states.
Both views have issues associated with them, as both contradict physical intuition in many cases, and neither has been conclusively proven to be the superior viewpoint.
The Spekkens toy model is designed to argue in favour of the epistemic viewpoint.
This model also implicitly assumes that there is an ontic state which the system is in at any given time, but simply that we are unable to observe it.
In particular, the model is one of local and noncontextual variables, which Bell's theorem tells us cannot ever reproduce all the predictions of quantum mechanics.
The Spekkens toy model is based on the knowledge balance principle "the number of questions about the physical state of a system that are answered must always be equal to the number that are unanswered in a state of maximal knowledge".
As such, in this case, the knowledge balance principle insists that the maximal number of questions in a canonical set that one can have answered at any given time is one, such that the amount of knowledge is equal to the amount of ignorance.
It is also assumed in the model that it is always possible to saturate the inequality, i.e. to have knowledge of the system exactly equal to that which is lacked, and thus at least two questions must be in the canonical set.
These are not allowed in a continuous model, however in this discrete system they arise as natural transformations.
There is, however, an analogy to a characteristically quantum phenomenon, that no allowed transformation functions as a universal state inverter.
This is due to the above fact, that a measurement can change the underlying ontic state of the system.
The nature of measurements and of the coherent superposition in this theory also gives rise to the quantum phenomenon of interference.
In the case of a two-system model, there is also a transformation that is analogous to the c-not operator on qubits.
Furthermore, within the bounds of the model it is possible to prove no-cloning and no-broadcasting theorems, reproducing a fair deal of the mechanics of quantum information theory.
[4] The toy model has also been analyzed from the viewpoint of categorical quantum mechanics.