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

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