# Bayesian uncertainty relation for a joint measurement of canonical variables

## TL;DR

This paper introduces a Bayesian uncertainty relation for joint measurements of canonical variables in quantum optics, allowing for practical assessment of measurement devices without the unbiasedness assumption.

## Contribution

It develops a Bayesian framework for joint-measurement uncertainty relations that applies to Gaussian states and does not require unbiasedness, enhancing experimental quantification.

## Key findings

- Reproduces the optimal bound for phase-space displacement estimation with pure Gaussian states
- Provides a practical method to evaluate joint-measurement devices using coherent states
- Offers a Bayesian approach that is experimentally feasible and theoretically robust

## Abstract

We present a joint-measurement uncertainty relation for a pair of mean square deviations of canonical variables averaged over Gaussian distributed quantum optical states. Our Bayesian formulation is free from the unbiasedness assumption, and enables us to quantify experimentally implemented joint-measurement devices by feeding a moderate set of coherent states. Our result also reproduces the most informative bound for quantum estimation of phase-space displacement in the case of pure Gaussian states.