RUI: Space of Multiparticle Entanglement Types

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Classification of entanglement types and describing the set of states whichshare a given entanglement type for quantum systems whose subsystems arequantum bits (qubits) is an important open problem in the field of quantuminformation theory. Its importance derives from applications including quantumcomputation algorithms and quantum communications protocols whichrely on entanglement as an essential resource.Most known results utilize polynomial invariants (polynomial functions ofstate vector coefficients) to analyze entanglement. The complexity of thisapproach grows exponentially with the number of qubits and thus far hasproved impractical to apply to systems with more than a small number ofqubits.An alternative approach using geometric invariants based on the Lie algebra of the group of local unitary transformations will be explored. In contrast to polynomial invariants, this is applicable to systems with an arbitrary number of qubits. The PIs have achieved general results that distinguish products of singletstates as having a unique extremal property among entanglement types.The PIs propose to further develop this machinery to refine existing invariants,to construct new invariants, and to prove a number of conjectures includingentanglement classification statements about product states and n-cat states.