Reconstruction of Gaussian Quantum States from Ideal Position Measurements: Beyond Pauli's Problem, I

TL;DR

This paper demonstrates that the covariance matrix of Gaussian quantum states can be reconstructed from position measurements using convex geometry concepts, enabling potential full state reconstruction in localized systems.

Contribution

It introduces a novel method leveraging polar duality and convex geometry to reconstruct covariance matrices of Gaussian states from position data.

Findings

Covariance matrices can be reconstructed from position measurements.

The method applies to all multidimensional Gaussian states.

The approach uses the John ellipsoid and quantum blobs for reconstruction.

Abstract

We show that the covariance matrix of a quantum state can be reconstructed from position measurements using the simple notion of polar duality, familiar from convex geometry. In particular, all multidimensional Gaussian states (pure or mixed) can in principle be reconstructed if the quantum system is well localized in configuration space. The main observation which makes this possible is that the John ellipsoid of the Cartesian product of the position localization by its polar dual contains a quantum blob, and can therefore be identified with the covariance ellipsoid of a quantum state.