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A genuinely $N$-partite entangled state may display vanishing $N$-partitecorrelations measured for arbitrary local observables. In such states thegenuine entanglement is noticeable solely in correlations between subsets ofparticles. A straightforward way to obtain such states for odd $N$ is to designan `anti-state' in which all correlations between an odd number of observersare exactly opposite. Evenly mixing a state with its anti-state then produces amixed state with no $N$-partite correlations, with many of them genuinelymultiparty entangled. Intriguingly, all known examples of `entanglement withoutcorrelations' involve an \emph{odd} number of particles. Here we furtherdevelop the idea of anti-states, thereby shedding light on the differentproperties of even and odd particle systems. We conjecture that there is noanti-state to any pure even-$N$-party entangled state making the simpleconstruction 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 evenviolate an $N$-partite Bell inequality, clearly demonstrating the non-classicalfeatures of these states as well as showing their applicability for quantumcommunication complexity tasks.

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