Spekkens Toy Model - Elementary Systems

Elementary Systems

For an elementary system, let 1 ∨ 2 represent the state of knowledge "The system is in the state one or the state 2." Under this model, there are six states of maximal knowledge that can be obtained: 1 ∨ 2, 1 ∨ 3, 1 ∨ 4, 2 ∨ 3, 2 ∨ 4 and 3 ∨ 4. There is also a single state less than maximal knowledge, corresponding to 1 ∨ 2 ∨ 3 ∨ 4. These can be mapped to six qubit states in a natural manner;

Failed to parse (Cannot store math image on filesystem.): 1 \lor 2 \iff | 0 \rangle

Failed to parse (Cannot store math image on filesystem.): 3 \lor 4 \iff | 1 \rangle

Failed to parse (Cannot store math image on filesystem.): 1 \lor 3 \iff | + \rangle

Failed to parse (Cannot store math image on filesystem.): 2 \lor 4 \iff | - \rangle

Failed to parse (Cannot store math image on filesystem.): 1 \lor 4 \iff | i \rangle

Failed to parse (Cannot store math image on filesystem.): 2 \lor 3 \iff | -i \rangle

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

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Under this mapping, it is clear that two states of knowledge in the toy theory correspond to two orthogonal states for the qubit if and only if they share no ontic states in common. This mapping also gives analogues in the toy model to quantum fidelity, compatibility, convex combinations of states and coherent superposition, and can be mapped to the Bloch sphere in the natural fashion. However, the analogy breaks down to a degree when considering coherent superposition, as one of the forms of the coherent superposition in the toy model returns a state which is orthogonal to what is expected with the corresponding superposition in the quantum model, and this can be shown to be an intrinsic difference between the two systems. This reinforces the earlier point that this model is not a restricted version of quantum mechanics, but instead a separate model which mimics quantum properties.