DMS Colloquium: Dr. Luke Oeding

Refreshments will be served in Parker 244, 3-3:25pm.

Speaker: Dr. Luke Oeding (Auburn)

Title: Nonlinear Algebra meets Tensors in Quantum Information

Abstract: Tensors represent quantum states, the core of quantum information (QI). Entanglement is the resource that drives the capabilities of quantum algorithms and the capacities of quantum channels. Two key problems guide our work: classify and separate distinct entanglement types and determine the states that have the most entanglement.

 Singular value decomposition (SVD) governs entanglement classification in the bipartite situation, and higher order singular value decomposition (HOSVD) plays a similar role for generic multi-partite states. In recent work [SIAGA 2025] with Ian Tan (Charles U., Prague) we extend HOSVD [De Lathauwer et al., SIMAX 2000] and provide algorithms for separating entanglement types for generic multi-qubit systems, generalizing work of Kraus [Phys Rev Let 2010] and others.

Classical invariant theory tells us that polynomial tensor invariants vanish on separable states, so the states that maximize invariants can be considered maximally entangled. Constrained optimization is the right tool to find these critical states, however, since tensor spaces grow exponentially, one also needs a toolkit that scales similarly. In another recent work [J. Physics A 2025] with Tan, we combine Vinberg theory for dimension reduction and homotopy continuation, which scales favorably with parallel computing, to solve this optimization problem. This method finds, in certain situations, all previously known entanglement maximizers.