Detecting and Quantifying Entanglement via Bayesian Updating

TL;DR

This paper introduces a Bayesian updating method to detect and quantify entanglement from limited measurement data, providing probabilistic assessments and error estimates without requiring complete state tomography.

Contribution

It presents a novel Bayesian approach that works with incomplete measurements and POVMs to detect and quantify entanglement, including error bars and volume estimates.

Findings

Successfully detects entanglement with incomplete measurements

Provides estimates of entanglement measures with error bars

Relates Bell inequality violations to entanglement measures

Abstract

We show how a straightforward Bayesian updating procedure allows one to detect and quantify entanglement from any finite set of measurement results. The measurements do not have to be tomographically complete, and may consist of POVMs rather than von Neumann measurements. One obtains a probability that one's state is entangled and an estimate of any desired entanglement measure, including their error bars. As an example we consider (tomographically incomplete) spin correlation measurements on both 2-qubit and 3-qubit states. As byproducts we obtain estimates of the volume of entangled states vs. states that violate a given Bell inequality for both pure and mixed states, and an inequality that relates the expectation value of the Bell operator to the negativity.