  It is proved that the width of a function and the width of the distribution
of its values cannot be made arbitrarily small simultaneously. In the case of
ergodic stochastic processes, an ensuing uncertainty relationship is
demonstrated for the product of correlation length and variance. A closely
related uncertainty principle is also established for the average degree of
fourth-order coherence and the spatial width of modes of bosonic quantum
fields. However, it is shown that, in the case of stochastic and quantum
observables, certain non-classical states with sub-Poissonian statistics, such
as for example photon number squeezed states in quantum optics, can overcome
the "classical" noise-resolution uncertainty limit. The described uncertainty
relationship, which is fundamentally different from the Heisenberg and related
uncertainty principles, can define an upper limit for the information capacity
of communication and imaging systems. It is expected to be useful in a variety
of problems in classical and quantum physics.
