Classification of entanglement types and describing the set of states which share a given entanglement type for quantum systems whose subsystems are quantum bits (qubits) is an important open problem in the field of quantum information theory. Its importance derives from applications including quantum computation algorithms and quantum communications protocols which rely on entanglement as an essential resource. Most known results utilize polynomial invariants (polynomial functions of state vector coefficients) to analyze entanglement. The complexity of this approach grows exponentially with the number of qubits and thus far has proved impractical to apply to systems with more than a small number of qubits. 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 singlet states 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 including entanglement classification statements about product states and n-cat states.
