Bounding entanglement dimensionality from the covariance matrix

TL;DR

This paper introduces a generalized covariance matrix criterion to certify high-dimensional entanglement and Schmidt number in bipartite quantum systems, especially useful in many-body experiments with limited measurement capabilities.

Contribution

It extends the Covariance Matrix Criterion to determine the Schmidt number, enabling detection of high-dimensional entanglement using collective observable variances.

Findings

Derived practical Schmidt-number criteria requiring minimal measurement data.

Showed criteria can detect a wider set of entangled states than fidelity-based methods.

Applied criteria to spin covariances relevant for cold atom experiments.

Abstract

High-dimensional entanglement has been identified as an important resource in quantum information processing, and also as a main obstacle for simulating quantum systems. Its certification is often difficult, and most widely used methods for experiments are based on fidelity measurements with respect to highly entangled states. Here, instead, we consider covariances of collective observables, as in the well-known Covariance Matrix Criterion (CMC)[1] and present a generalization of the CMC for determining the Schmidt number of a bipartite system. This is potentially particularly advantageous in many-body systems, such as cold atoms, where the set of practical measurements is very limited and only variances of collective operators can typically be estimated. To show the practical relevance of our results, we derive simpler Schmidt-number criteria that require similar information as the fidelity-based witnesses, yet can detect a wider set of states. We also consider paradigmatic criteria based on spin covariances, which would be very helpful for experimental detection of high-dimensional entanglement in cold atom systems. We conclude by discussing the applicability of our results to a multiparticle ensemble and some open questions for future work.