  One of the formulations of Heisenberg uncertainty principle, concerning
so-called measurement uncertainty, states that the measurement of one
observable modifies the statistics of the other. Here, we derive such a
measurement uncertainty principle from two comprehensible assumptions:
impossibility of instantaneous messaging at a distance (no-signaling), and
violation of Bell inequalities (non-locality). The uncertainty is established
for a pair of observables of one of two spatially separated systems that
exhibit non-local correlations. To this end, we introduce a gentle form of
measurement which acquires partial information about one of the observables. We
then bound disturbance of the remaining observables by the amount of
information gained from the gentle measurement, minus a correction depending on
the degree of non-locality. The obtained quantitative expression resembles the
quantum mechanical formulations, yet it is derived without the quantum
formalism and complements the known qualitative effect of disturbance implied
by non-locality and no-signaling.
