Quantum Tomographic Reconstruction with Error Bars: a Kalman Filter Approach

TL;DR

This paper introduces a Bayesian quantum tomography method using Kalman filters that provides optimal state estimates along with comprehensive uncertainty quantification, applicable to discrete measurement setups.

Contribution

It presents a novel Kalman filter-based approach for quantum state reconstruction that includes measurement uncertainties via covariance matrices, advancing towards a universal, statistically sound tomography method.

Findings

Successfully applied to real quantum optical experiments

Provides complete uncertainty quantification for reconstructed states

Lays groundwork for a versatile, error-aware tomography framework

Abstract

We present a novel quantum tomographic reconstruction method based on Bayesian inference via the Kalman filter update equations. The method not only yields the maximum likelihood/optimal Bayesian reconstruction, but also a covariance matrix expressing the measurement uncertainties in a complete way. From this covariance matrix the error bars on any derived quantity can be easily calculated. This is a first step towards the broader goal of devising an omnibus reconstruction method that could be adapted to any tomographic setup with little effort and that treats measurement uncertainties in a statistically well-founded way. In this first part we restrict ourselves to the important subclass of tomography based on measurements with discrete outcomes (as opposed to continuous ones), and we also ignore any measurement imperfections (dark counts, less than unit detector efficiency, etc.), which will be treated in a follow-up paper. We illustrate our general theory on real tomography experiments of quantum optical information processing elements.