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We propose a trade-off between the Lipschitz constants of the position andmomentum probability distributions for arbitrary quantum states. We refer tothe trade-off as a quantum reciprocity relation. The Lipschitz constant of afunction may be considered to quantify the extent of fluctuations of thatfunction, and is, in general, independent of its spread. The spreads of theposition and momentum distributions are used to obtain the celebratedHeisenberg quantum uncertainty relations. We find that the product of theLipschitz constants of position and momentum probability distributions isbounded below by a number that is of the order of the inverse square of thePlanck's constant.

Position and momentum cannot both be lazy: Quantum reciprocity relation with Lipschitz constants