Toward Jordan Decompositions of Tensors

Abstract

We expand on an idea of Vinberg to take a tensor space and the natural Lie algebra that acts on it and embed their direct sum into an auxiliary algebra. Viewed as endomorphisms of this algebra, we associate adjoint operators to tensors. We show that the group actions on the tensor space and on the adjoint operators are consistent, which means that the invariants of the adjoint operator of a tensor, such as the Jordan decomposition, are invariants of the tensor. We show that there is an essentially unique algebra structure that preserves the tensor structure and has a meaningful Jordan decomposition. We utilize aspects of these adjoint operators to study orbit separation and classification in examples relevant to tensor decomposition and quantum information.

arXiv commentTitle changed to reflect the revision. Extensive computations that will not appear in the published version are added to the appendix

References in corpus (18)

• Three qubits can be entangled in two inequivalent ways
• Four qubits can be entangled in nine different ways
• The Number of Eigenvalues of a Tensor
• Eigenvectors of tensors and algorithms for Waring decomposition
• Entanglement monotones and maximally entangled states in multipartite qubit systems
• Algebraic invariants of five qubits
• Entanglement of four qubit systems: a geometric atlas with polynomial compass I (the finite world)
• Geometric descriptions of entangled states by auxiliaries varieties
• Equations for the fifth secant variety of Segre products of projective spaces
• Entanglement of four-qubit systems: a geometric atlas with polynomial compass II (the tame world)
• Quantum Entanglement involved in Grover's and Shor's algorithms: the four-qubit case
• Singularity of type $D_4$ arising from four qubit systems
• A Spectral Theory for Tensors
• Invariant theory of ${\bigwedge }^3(9)$ and genus 2 curves
• Classification of four qubit states and their stabilisers under SLOCC operations
• Hyperdeterminants from the $E_8$ Discriminant
• Hecke cycles on moduli of vector bundles and orbital degeneracy loci
• Even nodal surfaces of K3 type