# Quantitative tomography for continuous variable quantum systems

## TL;DR

This paper introduces an efficient continuous variable quantum tomography method using Padua points for sampling, enabling accurate reconstruction of quantum states with fewer measurements and providing direct density matrix estimation with error bounds.

## Contribution

The paper presents a novel tomography scheme utilizing Padua points and Lagrange interpolation, significantly reducing measurement requirements and allowing direct density matrix estimation with error quantification.

## Key findings

- Reconstruction error decreases exponentially with the number of Padua points.
- The method achieves quasi-linear runtime in the number of sampling points.
- Density matrix elements can be estimated with linear error propagation and analytical bounds.

## Abstract

We present a continuous variable tomography scheme that reconstructs the Husimi Q-function (Wigner function) by Lagrange interpolation, using measurements of the Q-function (Wigner function) at the Padua points, the optimal sampling points for two dimensional reconstruction. Our approach drastically reduces the number of measurements required compared to using equidistant points on a regular grid, although reanalysis of such experiments is possible. The reconstruction algorithm produces a reconstructed function with exponentially decreasing error and quasi-linear runtime in the number of Padua points. Moreover, using the interpolating polynomial of the Q-function, we present a technique to directly estimate the density matrix elements of the continuous variable state, with only linear propagation of input measurement error. Furthermore, we derive a state-independent analytical bound on this error, such that our estimate of the density matrix is accompanied by a measure of its uncertainty.