Localizing multipartite entanglement with local and global measurements

Abstract

We study the task of localizing multipartite entanglement in pure quantum states onto a subsystem by measuring the remaining systems. To this end, we fix a multipartite entanglement measure and consider two quantities: the multipartite entanglement of assistance (MEA), defined as the entanglement measure averaged over the post-measurement states and maximized over arbitrary measurements; and the localizable multipartite entanglement (LME), defined in the same way but restricted to only local single-system measurements. We choose the n-tangle, the genuine multipartite entanglement concurrence and the concentratable entanglement (CE) as the underlying seed measure, and discuss the resulting MEA and LME quantities. First, we prove easily computable upper and lower bounds on MEA and LME and establish Lipschitz-continuity for the n-tangle and CE-based LME and MEA. Using these bounds we investigate the typical behavior of entanglement localization by deriving concentration inequalities for the MEA evaluated on Haar-random states and performing numerical studies for small tractable system sizes. We then turn our attention to protocols that transform graph states. We give a simple criterion based on a matrix equation to decide whether states with a specified n-tangle value can be obtained from a given graph state, providing no-go theorems for a broad class of such graph state transformations beyond the usual local Clifford plus local Pauli measurement framework. We generalize this analysis to weighted graph states and show that our entanglement localization framework certifies the near-optimality of recently discussed local-measurement protocols to transform uniformly weighted line graph states into GHZ states. Finally, we demonstrate how our MEA and LEA quantities can be used to detect phase transitions in transversal field Ising models.