  We show that a possible violation of the Robertson-Schr\"odinger uncertainty
principle may signal the existence of a deformation of the Heisenberg-Weyl
algebra. More precisely, we prove that any Gaussian in phase-space (even if it
violates the Robertson-Schr\"odinger uncertainty principle) is always a quantum
state of an appropriate non-commutative extension of quantum mechanics.
Conversely, all canonical non-commutative extensions of quantum mechanics
display states that violate the Robertson-Schr\"odinger uncertainty principle.
