Spekkens Toy Model - Elementary Systems - Transformations

Transformations

The only transformations on the ontic state of the system which respect the knowledge balance principle are permutations of the four ontic states. These map valid epistemic states to other valid epistemic states, for instance

Failed to parse (Cannot store math image on filesystem.): ((12)(34)) (1 \lor 2) \to 1 \lor 2

Failed to parse (Cannot store math image on filesystem.): ((12)(34)) (1 \lor 3) \to 2 \lor 4

Failed to parse (Cannot store math image on filesystem.): ((12)(3)(4)) (1 \lor 3) \to 2 \lor 3.

Considering again the analogy between the epistemic states of this model and the qubit states on the Bloch Sphere, these transformations consist of the typical allowed permutations of the six analogous states, as well as a set of permutations that are forbidden in the continuous qubit model. These are transformations such as (12)(3)(4) which correspond to antiunitary maps on Hilbert space. 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. In this case, this means that there is no single transformation S with the properties

Failed to parse (Cannot store math image on filesystem.): S(1 \lor 2) \to 3 \lor 4 \qquad S(3 \lor 4) \to 1 \lor 2

Failed to parse (Cannot store math image on filesystem.): S(1 \lor 3) \to 2 \lor 4 \qquad S(2 \lor 4) \to 1 \lor 3

Failed to parse (Cannot store math image on filesystem.): S(1 \lor 4) \to 2 \lor 3 \qquad S(2 \lor 3) \to 1 \lor 4.