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If no information gain implies no disturbance, then any discrete physical theory is classical

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(based on http://arxiv.org/abs/1210.0194)

It has been suggested
that nature could be discrete in the sense that the underlying state space of a
physical system has only a finite number of pure states. For example, the Bloch
ball of a single qubit could be discretized into small patches and only appear
round to us due to experimental limitations. Here, we present a strong physical
argument for the quantum theoretical property that every state space (even the
smallest possible one, the qubit) has infinitely many pure states. We propose a
simple physical postulate which dictates that in fact the only possible
discrete theory is classical mechanics. More specifically, we postulate that no
information gain implies no disturbance, or read in the contrapositive, that
disturbance leads to some form of information gain. In a theory like quantum mechanics
where we already know that the converse holds, i.e. information gain does imply
disturbance, this can be understood as postulating an equivalence between
disturbance and information gain. What is more, we show that non-classical
discrete theories are still ruled out even if we relax the postulate to hold
only approximately in the sense that no information gain only causes a small
amount of disturbance. Finally, our postulate also rules out popular
generalizations such as the PR-box that allows non-local correlations beyond
the limits of quantum theory.