2009/11/02 by Michael W. Hero, Hero, Michael W., Jeb F. Willenbring +3
Mathematics · #05C30 #22E70 #81P15 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Representation Theory (math.RT) #math.CO #math.RT #msc:05C30 #msc:22E70 #msc:81P15
paper · pdf · doi:10.48550/arxiv.0911.0222
Version to appear in the AMS Contemporary Mathematics Series
openalex publication_date 2009/11/02 · arxiv created 2011/10/28 · arxiv updated 2011/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We provide formulas for invariants defined on a tensor product of defining representations of unitary groups, under the action of the product group. This situation has a physical interpretation, as it is related to the quantum mechanical state space of a multi-particle system in which each particle has finitely many outcomes upon observation. Moreover, these invariant functions separate the entangled and unentangled states, and are therefore viewed as measurements of quantum entanglement. When the ranks of the unitary groups are large, we provide a graph theoretic interpretation for the dimension of the invariants of a fixed degree. We also exhibit a bijection between isomorphism classes of finite coverings of connected simple graphs and a basis for the space of invariants. The graph coverings are related to branched coverings of surfaces.