Position-Momentum Uncertainty Relations in the Presence of Quantum Memory

TL;DR

This paper extends entropic uncertainty relations to include position and momentum measurements with quantum memory, demonstrating implications for quantum cryptography and security in continuous-variable systems.

Contribution

It introduces entropic uncertainty relations for observables with infinite or continuous spectra considering quantum memory, advancing the understanding of quantum uncertainty in continuous-variable systems.

Findings

Uncertainty relations derived for position and momentum with quantum memory.

Implication for security in quantum key distribution using Gaussian states.

Operational relevance in quantum cryptography with continuous variables.

Abstract

A prominent formulation of the uncertainty principle identifies the fundamental quantum feature that no particle may be prepared with certain outcomes for both position and momentum measurements. Often the statistical uncertainties are thereby measured in terms of entropies providing a clear operational interpretation in information theory and cryptography. Recently, entropic uncertainty relations have been used to show that the uncertainty can be reduced in the presence of entanglement and to prove security of quantum cryptographic tasks. However, much of this recent progress has been focused on observables with only a finite number of outcomes not including Heisenberg's original setting of position and momentum observables. Here we show entropic uncertainty relations for general observables with discrete but infinite or continuous spectrum that take into account the power of an entangled observer. As an illustration, we evaluate the uncertainty relations for position and momentum measurements, which is operationally significant in that it implies security of a quantum key distribution scheme based on homodyne detection of squeezed Gaussian states.