Characterizing Entanglement Dimensionality from Randomized Measurements

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 da = db = 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.

Automated summary

We propose a method to detect the dimensionality of entanglement using correlations between measurements in randomized directions. By deriving an inequality based on the covariance matrix criterion, which is invariant under local changes of su(d) bases, we can find regions in the space of randomized correlations moments that determine the different dimensionalities of entanglement. Our method shows promising results in practical scenarios and can detect more states than existing criteria, making it a powerful and potentially simpler approach. Future work should focus on implementing this method in multipartite scenarios.