We present methods for detecting entanglement around N-qubit Dicke states with N/2 excitations. In the first part of the paper we show that for large N these states have the smallest overlap possible with states without multi-partite entanglement. Thus these states are particulary well suited for experiments aimed to create and detect genuine multi-partite entanglement. We present fidelity-based entanglement witness operators for detecting multipartite entanglement around these states. In the second part of the paper we consider entanglement criteria, somewhat similar to spin squeezing, based on the moments or variances of the collective spin operators. We present both criteria detecting entanglement in general and criteria detecting only genuine multi-partite entanglement.
The detection of multipartite entanglement in multipartite quantum systems is a fundamental and key issue in quantum information theory.In this paper, we investigate
k-nonseparability and
k-partite entanglement of
N-partitequantum systems from the perspective of the generalized Wigner-Yanase skew information introduced by Yang
et al.[\href{https://doi.org/10.1103/PhysRevA.106.052401}{Phys. Rev. A \textbf{106}, 052401 (2022)}]. More specifically, we develop two different approaches in form of inequalities to construct entanglement criteria, which are expressed in terms of the generalized Wigner-Yanase skew information.Any violation of these inequalities by a quantum state reveals its
k-nonseparability or
k-partite entanglement,so these inequalities present the hierarchic classifications of
k-nonseparability or
k-partite entanglement for all
N-partite quantum states from
N-nonseparability to
2-nonseparability or from
2-partite entanglement to
N-partite entanglement, which are more refined than well-known ways. It is shown that our results reveal some
k-nonseparability and
k-partite entanglement that remain undetected by other methods,and these are illustrated through some examples.
We present a method to detect entanglement partitions of multipartite quantum systems, by exploiting their inherent symmetries. Structures like genuinely multipartite entanglement,
m-separability and entanglement depth are detected as very special cases. This formulation enables us to characterize all the entanglement partitions of all three- and four- partite states and witnesses with unitary and permutation symmetry. In particular, we find and parametrize a complete set of bound entangled states therein. For larger systems, we provide a large family of analytical witnesses detecting many-body states of arbitrary size where none of the parties is separable from the rest. This method relies on weak Schur sampling with projective measurements, and thus can be implemented in a quantum computer. Beyond physics, our results extend to the mathematical literature: we establish new inequalities between matrix immanants, and characterize the set of such inequalities for matrices of size three and four.