Interpreting Multipartite Entanglement through Topological Summaries

TL;DR

This paper links topological summaries from TDA to operational measures of multipartite entanglement, providing bounds and characterizations that enhance understanding and verification of complex entangled states.

Contribution

It introduces the first operational interpretations of topological quantities in multipartite entanglement, connecting them to entanglement measures and state verification methods.

Findings

Bound on Integrated Euler Characteristic via distillable entanglement

Characterization of GHZ states through entanglement complex topology

Potential new verification schemes for highly entangled states

Abstract

The study of multipartite entanglement is much less developed than the bipartite scenario. Recently, a string of results have proposed using tools from Topological Data Analysis (TDA) to attach topological quantities to multipartite states. However, these quantities are not directly connected to concrete information processing tasks making their interpretations vague. We take the first steps in connecting these abstract topological quantities to operational interpretations of entanglement in two scenarios. The first is we provide a bound on the Integrated Euler Characteristic defined by Hamilton and Leditzky via an average distillable entanglement, which we develop as a generalization of the Meyer-Wallach entanglement measure studied by A. J. Scott in 2004. This allows us to connect the distance of an error correcting code to the Integrated Euler Characteristic. The second is we provide a characterization of a class of graph states containing the GHZ state via the birth and death times of the connected components and 1-dimensional cycles of the entanglement complex. In other words, the entanglement distance behavior of the first Betti number $\beta_1(\varepsilon)$ allows us to determine if a state is locally equivalent to a GHZ state, potentially providing new verification schemes for highly entangled states.