  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.
