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

TL;DR

This paper introduces a quantum reciprocity relation linking the Lipschitz constants of position and momentum distributions, revealing a fundamental trade-off that complements the Heisenberg uncertainty principle.

Contribution

It establishes a novel trade-off between Lipschitz constants of quantum distributions, providing a new perspective on quantum fluctuations and their relation to uncertainty.

Findings

Product of Lipschitz constants bounded below by inverse square of Planck's constant

Lipschitz constants quantify fluctuations independently of spread

Trade-off complements traditional uncertainty relations

Abstract

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