DC 11 – Geometry of Hermitian tensor spaces

Project Title: Geometry of Hermitian tensor spaces
Advisors: G. Ottaviani (UniFI), Yang Qi (Inria), B. Mourrain (Inria) Mentor: E. Rubei (UniFI)
Objectives: Tensor spaces have natural distances invariant under the orthogonal group or the unitary group, respectively in the real or the complex setting. Several optimisation problems in tensor spaces consist in minimizing the distance between a given tensor and a orthogonally (or unitary) invariant subvariety. Best rank approximation is an important meaningful case and other important examples come from Quantum Information, where the tensor space is the Qubit Space. These problems are in general highly non-convex, and analyzing the corresponding landscapes is particularly interesting and helpful. This motivates us to study the critical points of the above distance functions. It was observed in the orthogonal setting that the complexity of this optimisation problem is often proportional to the number of complex critical points of the corresponding distance function, which is called Euclidean Distance degree. The goal is to develop techniques to compute the Hermitian Distance degree (seen as analog of Euclidean Distance degree), which is expected to be constant along real chambers. Basic questions are: How many different values it may have, can the chambers and the boundaries among them be characterized ? What are the values for rank-one tensors ? As a special case, in Qubit Space, this study may contribute to the geometric measure of Entanglement.
Expected Results: New results on the Hermitian Distance degree, on the real chambers, where it is constant, on the geometric measure of Entanglement. New applications to Qubit Spaces. A journal paper in each year and completion of a PhD degree by the DC.
Planned secondment(s): with B. Mourrain and Yang Qi at Inria (M25-29, M37-41), to study the topology of real chambers where Hermitian Distance degree is constant and to compute numerically and symbolically basic examples in Tensor Spaces.
Joint degree: University of Florence, Université Côte d’Azur

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