It is shown that 'non-quantum systems', with anomalous statistical properties, would carry a distinctive experimental signature. Such systems can exist in deterministic hidden-variables theories (such as the pilot-wave theory of de Broglie and Bohm). The signature consists of non-additive expectations for non-commuting observables, breaking the sinusoidal modulation of quantum probabilities for two-state systems. This effect is independent of the quantum state (pure or mixed), or of the details of the hidden-variables model. Experiments are proposed, testing polarisation probabilities for single photons.

11 pages, Latex, no figures