Great discussion starter, Jonathan. If I may:
Conventional quantum mechanics is probabilistic. In the absence of any information of a state, the probability is 50-50 for the switch to be on or off. You (and Schrodinger's cat) are either alive or dead. There is no betweeness; the odds are equally likely, in any moment, that you are breathing or not.
This "equally likely" hypothesis that rules probability analysis applies regardless of how many numerical dimensions are in play. E.g., a six sided die will have 3 0s and 3 1s. A fair die so marked, like a fair 2-sided coin, will stop on one value 50% of the time in a sufficiently long run of Bernoulli trials. The upper bound for one result or another is 0.5.
As Joy explained early on: " ... on a 3-sphere we can always find three linearly-independent vector fields that are nowhere vanishing."
This is critical information, because we go from the probabilistic odds of having no information, that forces us to apply the "equally likely" hypothesis with no measure space in mind -- to a space of perfect information, where we are certain of analytic continuation of a particle pair measure that is always correlated continuously from a freely chosen initial condition. The framework is coordinate free, which is why the vector fields are linearly independent.
The upper bound of 2 derives from 2/4 = +/- 0.5. Joy's upper bound derives from 2\/2/4 = +/- 0.7071 ... a continuous function that returns to its starting value only by 4pi rotation vice the 2pi rotation of linearly dependent values in superposition.
Most are interested in experimental results. I have been interested in the mathematics, ever since I first realized years ago that the framework is generally covariant and thus a quantum bridge to general relativity.
