Spekkens Toy Model - The Model

The Model

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." However, the "knowledge" one can possess about a system must be carefully defined for this principle to have any meaning. To do this, the concept of a canonical set of yes or no questions is defined as the minimum number of questions needed. For example, for a system with 4 states, one can ask "Is the system in state 1?", "Is the system in state 2?" and "Is the system in state 3?" which would determine the state of the system (state 4 being the case if all three questions were answered "No."). However, one could also ask "Is the system in either state 1 or state 2?" and "Is the system in either state 1 or state 3?", which would also uniquely determine the state, and has only two questions in the set. This set of questions is not unique, however it is clear that at least two questions (bits) are required to exactly represent one of four states. We say that for a system with 4 states, the number of questions in a canonical set is two. As such, in this case, the knowledge balance principle insists that the maximum 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. Since no question is allowed to exactly specify the state of the system, the number of possible ontic states must be at least 4 (if it were less than 4, the model would be trivial, since any question that could be asked may return an answer specifying the exact state of the system, thus no question can be asked). Since a system with four states (described above) exists, it is referred to as an elementary system. The model then also assumes that every system is built out of these elementary systems, and that each subsystem of any system also obeys the knowledge balance principle.