Entanglement Structure and Matrix Inequalities from Isotypic Measurements

We 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 entanglement partitions of all three- and four-partite states and witnesses with unitary and permutation symmetry, which we denote Schur-Weyl isotypic witnesses. 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 multipartite states of arbitrary size where none of the parties is separable from the rest. The proposed method relies on weak Schur sampling with projective measurements onto isotypic components and can be implemented in a quantum computer. Beyond physics, our results extend to the mathematical literature: we establish new inequalities between matrix immanants, which constitute an open problem, and characterize the set of such inequalities for matrices of sizes three and four.