Pearson Correlation Coefficient as a measure for Certifying and Quantifying High Dimensional Entanglement

TL;DR

This paper evaluates the effectiveness of using Pearson correlation coefficient (PCC) to certify and quantify high-dimensional entanglement in bipartite quantum states, extending previous qubit-based methods to qutrits and higher dimensions.

Contribution

It provides a comprehensive analysis of PCC-based entanglement characterization for bipartite qutrit states and explores its extension to higher dimensions d=4 and 5.

Findings

PCC can effectively certify entanglement in bipartite qutrit states.

The scheme quantifies entanglement with limited measurements.

Extension to higher dimensions is feasible and discussed.

Abstract

A scheme for characterizing entanglement using the statistical measure of correlation given by the Pearson correlation coefficient (PCC) was recently suggested that has remained unexplored beyond the qubit case. Towards the application of this scheme for the high dimensional states, a key step has been taken in a very recent work by experimentally determining PCC and analytically relating it to Negativity for quantifying entanglement of the empirically produced bipartite pure state of spatially correlated photonic qutrits. Motivated by this work, we present here a comprehensive study of the efficacy of such an entanglement characterizing scheme for a range of bipartite qutrit states by considering suitable combinations of PCCs based on a limited number of measurements. For this purpose, we investigate the issue of necessary and sufficient certification together with quantification of entanglement for the two-qutrit states comprising maximally entangled state mixed with white noise and coloured noise in two different forms respectively. Further, by considering these classes of states for d=4 and 5, extension of this PCC based approach for higher dimensions d is discussed.