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.
Abstract A detailed investigation of the multiparty entanglement present in the 4 − qubit quantum hypergraph states is presented, following a measurement‐based geometrical approach. Considering a classification of the 4 − party quantum system represented by a mathematical hypergraph based on the connections between its vertices, the genuine 4 − party entanglement present in each bi‐partition of the states have been measured. A strong correlation between the connectivity of the vertices of the hypergraphs and the genuine 4 − party entanglement has been found. The equivalence of the genuine 4 − party entanglement present in each bi‐partition is shown considering similar connectivity of the vertices. This explicates the cyclic permutation symmetry of the multiparty entanglement present in the 4 − qubit hypergraph states. Physically, one may expect the quantum systems with superposition of many states to behave in this symmetric manner while mapped into a network‐type picture, which the authors have quantified, as well as classified in this work.
We investigate the structural aspects of genuine multipartite entanglement in Gaussian states of continuous variable systems. Generalizing the results of Adesso and Illuminati [Phys. Rev. Lett. 99, 150501 (2007)], we analyze whether the entanglement shared by blocks of modes distributes according to a strong monogamy law. This property, once established, allows us to quantify the genuine
N-partite entanglement not encoded into $2,\dots{},K,\dots{},(N\ensuremath{-}1)$-partite quantum correlations. Strong monogamy is numerically verified, and the explicit expression of the measure of residual genuine multipartite entanglement is analytically derived, by a recursive formula, for a subclass of Gaussian states. These are fully symmetric (permutation-invariant) states that are multipartitioned into blocks, each consisting of an arbitrarily assigned number of modes. We compute the genuine multipartite entanglement shared by the blocks of modes and investigate its scaling properties with the number and size of the blocks, the total number of modes, the global mixedness of the state, and the squeezed resources needed for state engineering. To achieve the exact computation of the block entanglement, we introduce and prove a general result of symplectic analysis: Correlations among
K blocks in
N-mode multisymmetric and multipartite Gaussian states, which are locally invariant under permutation of modes within each block, can be transformed by a local (with respect to the partition) unitary operation into correlations shared by
K single modes, one per block, in effective nonsymmetric states where $N\ensuremath{-}K$ modes are completely uncorrelated. Due to this theorem, the above results, such as the derivation of the explicit expression for the residual multipartite entanglement, its nonnegativity, and its scaling properties, extend to the subclass of non-symmetric Gaussian states that are obtained by the unitary localization of the multipartite entanglement of symmetric states. These findings provide strong numerical evidence that the distributed Gaussian entanglement is strongly monogamous under and possibly beyond specific symmetry constraints, and that the residual continuous-variable tangle is a proper measure of genuine multipartite entanglement for permutation-invariant Gaussian states under any multipartition of the modes.