The hyperlink representation of entanglement and the inclusion-exclusion principle

TL;DR

This paper introduces entanglement hyperlinks (EHLs), a generalized measure based on the inclusion-exclusion principle, providing an exact way to characterize multipartite entanglement in quantum states.

Contribution

The work extends entanglement links to exact entanglement hyperlinks, enabling precise decomposition of multipartite entanglement using the inclusion-exclusion principle.

Findings

EHLs crossing a factorized partition vanish

EHLs between blocks can be expressed as sums of connecting EHLs

EHLs offer a detailed characterization of multipartite entanglement

Abstract

The entanglement entropy (EE) of any bipartition of a pure state can be approximately expressed as a sum of entanglement links (ELs). In this work, we introduce their exact extension, i.e. the entanglement hyperlinks (EHLs), a type of generalized mutual informations defined through the inclusion-exclusion principle, each of which captures contributions to the multipartite entanglement that are not reducible to lower-order terms. We show that any EHL crossing a factorized partition must vanish, and that the EHLs between any set of blocks can be expressed as a sum of all the EHLs that join all of them. This last result allows us to provide an exact representation of the EE of any block of a pure state, from the sum of the EHLs which cross its boundary. In order to illustrate their rich structure, we discuss some explicit numerical examples using ground states of local Hamiltonians. The EHLs thus provide a remarkable tool to characterize multipartite entanglement in quantum information theory and quantum many-body physics.