Genuinely entangled symmetric states with no $N$-partite correlations

TL;DR

This paper characterizes symmetric genuinely entangled N-qubit states lacking N-partite correlations, revealing their non-existence for even N and providing explicit constructions for odd N, especially for three qubits and rank-2 states.

Contribution

It offers a simple tensor-based characterization of such states, proves their non-existence for even N, and constructs families for odd N, advancing understanding of multipartite entanglement.

Findings

Symmetric states with no N-partite correlations cannot exist for even N.

Complete characterization of such states for three qubits.

Construction method for states with odd N, including rank-2 states.

Abstract

We investigate genuinely entangled $N$-qubit states with no $N$-partite correlations in the case of symmetric states. Using a tensor representation for mixed symmetric states, we obtain a simple characterization of the absence of $N$-partite correlations. We show that symmetric states with no $N$-partite correlations cannot exist for an even number of qubits. We fully identify the set of genuinely entangled symmetric states with no $N$-partite correlations in the case of three qubits, and in the case of rank-2 states. We present a general procedure to construct families for an arbitrary odd number of qubits.