The Optimal Pure Gaussian State Associated with Joint Position-Momentum Measurements

TL;DR

This paper introduces a method to construct the optimal pure Gaussian state from joint position-momentum measurement data using multivariate statistical analysis and symplectic geometry, providing a practical approach for quantum state approximation.

Contribution

It presents a novel technique combining statistical and symplectic tools to derive the best pure Gaussian state approximation from experimental data.

Findings

Successfully constructs optimal Gaussian states from measurement data.

Integrates multivariate analysis with quantum uncertainty principles.

Provides a practical framework for quantum state estimation.

Abstract

We show that given experimental data obtained from joint position and momentum measurements one can construct an optimal pure state approximating the observed quantum state. For that purpose we use a tool from multivariate statistical analysis (the MVE method), which relies on the existence of a minimum volume ellipsoid (the John--L\"owner ellipsoid) containing a given convex set. This method allows us to determine the shape of the covariance ellipsoid, which is thereafter calibrated using the expression of the uncertainty principle in terms of the notion of symplectic capacity. We finally use the Wigner formalism to produce the best approximating Gaussian state.