Research

Most of my research revolves around tensors and their applications in quantum information, physics and data science. I am mostly working at the mathematics level of rigor, but enjoy working with physicists from time to time at a more heuristic level. Most of my work uses (free) probability theory, random tensors and matrices.

Under construction…. (more to come soon!)

Tensor norms, spin glasses and entanglement

Tensor norms provide measure of entanglement. In the multipartite case, these measures are not fully understood and part of my research focuses on understanding those better.

A prototypical example of a tensor norm, of use in quantum information theory, is the injective norm¹ or equivalently, the geometric entanglement.

Given a pure state $\vert \psi \rangle\in \mathbb{C}^{d_1}\otimes \ldots \otimes \mathbb{C}^{d_p}$ , its injective norm is its overlap with the closest separable state that is $\max_{x_i}\vert \langle \psi\vert x_1\ldots x_p\rangle\vert$ .

There lies the analogy with spherical spin glasses. In fact, for generic quantum states computing their injective norm is a NP-hard problem. Note that $\vert \psi \rangle$ components can be seen as the complex couplings of a spherical spin glass Hamiltonian

$H=\sum_{i_1...i_p}\psi_{i_1...i_p}(x_1)_{i_1}...(x_p)_{i_p}$

Determining the injective norm becomes the statistical physics problem of computing (minus) the ground state of $H$ .

I currently work on understanding:

Typical values of such norms for different ensembles of random states and tensors – the connection with spin glass theory and statistical physics is strenghten
Extracting those norms values for interesting family of states and tensors
Building new entanglement measures from related objects

Tensors and data analysis

Under construction…. (more to come soon!)

Actually there is a family of injective norms, since those are induced. ↩︎