# Measurement uncertainty relations for position and momentum: Relative entropy formulation

## TL;DR

This paper establishes a lower bound on the information loss, measured by relative entropy, when approximating joint measurements of position and momentum observables in quantum systems, focusing on Gaussian states and measurements.

## Contribution

It introduces a new entropic error function to quantify measurement uncertainty and derives bounds for position-momentum observables in various configurations.

## Key findings

- Derived a lower bound for information loss in joint measurements
- Connected quantum bounds to the dimension of the system
- Showed transition from incompatible to compatible observables based on measurement directions

## Abstract

Heisenberg's uncertainty principle has recently led to general measurement uncertainty relations for quantum systems: incompatible observables can be measured jointly or in sequence only with some unavoidable approximation, which can be quantified in various ways. The relative entropy is the natural theoretical quantifier of the information loss when a `true' probability distribution is replaced by an approximating one. In this paper, we provide a lower bound for the amount of information that is lost by replacing the distributions of the sharp position and momentum observables, as they could be obtained with two separate experiments, by the marginals of any smeared joint measurement. The bound is obtained by introducing an entropic error function, and optimizing it over a suitable class of covariant approximate joint measurements. We fully exploit two cases of target observables: (1) $n$-dimensional position and momentum vectors; (2) two components of position and momentum along different directions. In (1), we connect the quantum bound to the dimension $n$; in (2), going from parallel to orthogonal directions, we show the transition from highly incompatible observables to compatible ones. For simplicity, we develop the theory only for Gaussian states and measurements.