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Entanglement Structure and Matrix Inequalities from Isotypic Measurements

Albert Rico1, Dmitry Grinko2,3,4, Robin Krebs5, and Lin Htoo Zaw6

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Phys. Rev. Lett. 137, 100203 – Published 4 September, 2026

DOI: https://doi.org/10.1103/nvk2-h8d5

Abstract

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.

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Phys. Rev. Lett. 137, 100203– Published 4 September, 2026
  • Received 7 February 2026
  • Revised 27 July 2026
  • Accepted 3 August 2026
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