# Genuine $N$-partite entanglement without $N$-partite correlation functions

## TL;DR

This paper explores states of multiple particles that are genuinely entangled but lack $N$-partite correlations, revealing differences between even and odd particle systems and demonstrating their non-classical properties and potential for quantum communication.

## Contribution

It introduces higher-rank examples of entanglement without correlations for even $N$, and analyzes the properties and limitations of anti-states in such systems.

## Key findings

- Odd $N$ states can be constructed with anti-states to eliminate $N$-partite correlations.
- Higher-rank states for even $N$ exist and are genuinely entangled.
- These states can violate $N$-partite Bell inequalities, showing non-classicality.

## Abstract

A genuinely $N$-partite entangled state may display vanishing $N$-partite correlations measured for arbitrary local observables. In such states the genuine entanglement is noticeable solely in correlations between subsets of particles. A straightforward way to obtain such states for odd $N$ is to design an `anti-state' in which all correlations between an odd number of observers are exactly opposite. Evenly mixing a state with its anti-state then produces a mixed state with no $N$-partite correlations, with many of them genuinely multiparty entangled. Intriguingly, all known examples of `entanglement without correlations' involve an \emph{odd} number of particles. Here we further develop the idea of anti-states, thereby shedding light on the different properties of even and odd particle systems. We conjecture that there is no anti-state to any pure even-$N$-party entangled state making the simple construction scheme unfeasable. However, as we prove by construction, higher-rank examples of `entanglement without correlations' for arbitrary even $N$ indeed exist. These classes of states exhibit genuine entanglement and even violate an $N$-partite Bell inequality, clearly demonstrating the non-classical features of these states as well as showing their applicability for quantum communication complexity tasks.