Knowledge and ignorance in incomplete quantum state tomography

TL;DR

This paper introduces a quantum state reconstruction method that maximizes both likelihood and entropy to select the least-biased estimator consistent with incomplete measurement data, balancing knowledge and ignorance.

Contribution

It proposes a novel reconstruction scheme combining likelihood and entropy maximization for incomplete quantum state tomography, addressing non-uniqueness issues.

Findings

The method yields a unique, least-biased estimator.

It provides a systematic way to incorporate ignorance into quantum state reconstruction.

The approach reveals new estimator structures in incomplete tomography.

Abstract

Quantum state reconstruction on a finite number of copies of a quantum system with informationally incomplete measurements does, as a rule, not yield a unique result. We derive a reconstruction scheme where both the likelihood and the von Neumann entropy functionals are maximized in order to systematically select the most-likely estimator with the largest entropy, that is the least-bias estimator, consistent with a given set of measurement data. This is equivalent to the joint consideration of our partial knowledge and ignorance about the ensemble to reconstruct its identity. An interesting structure of such estimators will also be explored.