Negativity vs. Purity and Entropy in Witnessing Entanglement

TL;DR

This paper compares measures of mixedness like purity and entropy to witness entanglement, finding that purity-based measures outperform entropy-based ones at high dimensions, with implications for understanding quantum correlations.

Contribution

It demonstrates that purity measures are more effective than entropy measures in witnessing high-dimensional entanglement, offering a resource-efficient approach.

Findings

Purity comparisons outperform entropy in high-dimensional entanglement detection

All entangled states have a negative partial transpose (NPT)

Purity-based methods require fewer resources than entropy-based methods

Abstract

In this work, we show that while all measures of mixedness may be used to witness entanglement, all such entangled states must have a negative partial transpose (NPT). Though computing the negativity of the partial transpose scales well at high dimension, it relies on knowing the complete quantum state, which does not. To address this, we compare different measures of mixedness over uniform ensembles of joint quantum states at varying dimension to gauge their relative success in witnessing entanglement. In doing so, we find that comparing joint and marginal purities is overwhelmingly more successful at high dimension at identifying entanglement than comparing joint and marginal von Neumann entropies, in spite of requiring fewer resources. We conclude by showing how our results impact the fundamental relationship between correlation and entanglement and related witnesses.