Groups of Elementary Systems
A pair of elementary systems has 16 combined ontic states, corresponding to the combinations of the numbers 1 through 4 with 1 through 4 (i.e. the system can be in the state (1,1), (1,2), etc.) The epistemic state of the system is limited by the knowledge balance principle once again. Now however, not only does it restrict the knowledge of the system as a whole, but also of both of the constituent subsystems. Two types of systems of maximal knowledge arise as a result. The first of these corresponds to having maximal knowledge of both subsystems; for example, that the first subsystem is in the state 1 ∨ 3 and the second is in the state 3 ∨ 4, meaning that the system as a whole is in one of the states (1,3), (1,4), (3,3) or (3,4). In this case, nothing is known about the correspondence between the two systems. The second is more interesting, corresponding to having no knowledge about either system individually, but having maximal knowledge about their interaction. For example, one could know that the ontic state of the system is one of (1,1), (2,2), (3,4) or (4,3). Here nothing is known about the state of either individual system, but knowledge of one system gives knowledge of the other. This corresponds to the entangling of particles in quantum theory.
It is possible to consider valid transformations on the states of a group of elementary systems, although the mathematics of such an analysis is more complicated than the case for a single system. Transformations consisting of a valid transformation on each state acting independently are always valid. In the case of a two system model, there is also a transformation which 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.
The monogamy of pure entanglement also has a strong analogue within the toy model, as a group of three or more systems in which knowledge of one system would grant knowledge of the others would break the knowledge balance principle. An analogy of quantum teleportation also exists in the model, as well as a number of important quantum phenomena.
Read more about this topic: Spekkens Toy Model
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