Quantum State Reconstruction From Incomplete Data

TL;DR

This paper explores methods for estimating quantum states from incomplete data, comparing approaches like MaxEnt, Bayesian inference, and optimal measurements to improve state reconstruction accuracy.

Contribution

It introduces and analyzes multiple schemes for quantum state reconstruction from limited measurement data, including MaxEnt, Bayesian, and optimal measurement strategies.

Findings

MaxEnt efficiently estimates states on incomplete data

Bayesian inference reconstructs states from finite samples

Optimal measurements maximize fidelity in state estimation

Abstract

Knowing and guessing, these are two essential epistemological pillars in the theory of quantum-mechanical measurement. As formulated quantum mechanics is a statistical theory. In general, a priori unknown states can be completely determined only when measurements on infinite ensembles of identically prepared quantum systems are performed. But how one can estimate (guess) quantum state when just incomplete data are available (known)? What is the most reliable estimation based on a given measured data? What is the optimal measurement providing only a finite number of identically prepared quantum objects are available? These are some of the questions we address. We present several schemes for a reconstruction of states of quantum systems from measured data: (1) We show how the maximum entropy (MaxEnt) principle can be efficiently used for an estimation of quantum states on incomplete observation levels. (2) We show how Bayesian inference can be used for reconstruction of quantum states when only a finite number of identically prepared systems are measured. (3) We describe the optimal generalized measurement of a finite number of identically prepared quantum systems which results in the estimation of a quantum state with the highest fidelity.