Characterizing entanglement dimensionality from randomized measurements

TL;DR

This paper introduces a new inequality-based criterion for detecting the dimensionality of entanglement using randomized measurement correlations, which is invariant under local basis changes and effective in practical scenarios.

Contribution

We derive a novel entanglement dimensionality criterion based on randomized measurements, extending previous results and providing analytical boundaries for various dimensions.

Findings

The criterion detects higher-dimensional entanglement more effectively than existing methods.

Analytical boundary curves for entanglement dimensionalities are established for bipartite systems.

The method remains effective with finite statistical samples and is potentially simpler to implement.

Abstract

We consider the problem of detecting the dimensionality of entanglement with the use of correlations between measurements in randomized directions. First, exploiting the recently derived covariance matrix criterion for the entanglement dimensionality [S. Liu et al., arXiv:2208.04909], we derive an inequality that resembles well-known entanglement criteria, but contains different bounds for the different dimensionalities of entanglement. This criterion is invariant under local changes of $su(d)$ bases and can be used to find regions in the space of moments of randomized correlations, generalizing the results of [S. Imai et al., Phys. Rev. Lett. 126, 150501 (2021)] to the case of entanglement-dimensionality detection. In particular, we find analytical boundary curves for the different entanglement dimensionalities in the space of second- and fourth-order moments of randomized correlations for all dimensions $d_a = d_b = d$ of a bipartite system. We then show how our method works in practice, also considering a finite statistical sample of correlations, and we also show that it can detect more states than other entanglement-dimensionality criteria available in the literature, thus providing a method that is both very powerful and potentially simpler in practical scenarios. We conclude by discussing the partly open problem of the implementation of our method in the multipartite scenario.