DC 12 – Approximation hierarchies for quantum entanglement detection

Project Title: Approximation hierarchies for quantum entanglement detection
Advisors: M. Laurent (NWO-I), V. Magron (CNRS), Mentor: M. Almeida (Quantinuum)
Objectives: Tensor products model quantum state systems consisting of multiple subsystems. Detecting whether a quantum state is entangled (or separable) is a fundamental question in quantum information theory, which boils down to studying density matrices on tensor products of Hilbert spaces. One needs to develop tractable approximations for tackling this computationally hard problem. Tight links have been recently established between an approximation hierarchy of Doherty, Parrilo and Spedaglieri for separable states (based on state extension) and the classical moment and sum-of-squares approach for polynomial optimisation. In this project we will investigate further methods from conic and polynomial optimisation, aiming to design dedicated tools for some well-structured classes of (bipartite) quantum states. This includes the states that satisfy invariance properties under certain group actions, which were recently characterized as arising from (pairwise) completely positive matrices. The cone of completely positive matrices is well-studied in optimisation and algebraic matrix theory, but the more general cone of pairwise completely positive matrices much less. A natural question is developing approximation hierarchies for these cones and investigating how they can be used to design better approximations for the corresponding quantum states. Understanding the links between the ranks of completely positive factorizations and the separable rank of quantum states is also a relevant question in this context. Hence, this project revolves around the interplay between conic approximations for matrix properties and their use for a better understanding of quantum entanglement for well-structured bipartite quantum states. Exploiting the structure of the initial quantum information problem, such as correlative or term sparsity will be decisive to improve the scalability of the related hierarchies of conic programming formulations.
Expected Results: Develop methods for better approximation hierarchies of separable quantum states and more efficient entanglement witnesses, and investigate the interplay between various classical matrix properties and their use for a better understanding of quantum entanglement. Complete a journal paper in each year and completion of a PhD degree by the DC.
Planned secondment(s): The DC will have a 10-month secondment at LAAS-CNRS (M21-30) to collaborate with the second advisor and learn about exploiting structure in conic and polynomial optimisation for efficient implementation of the developed methods. A 2-month secondment is planned at Quantinuum (M31-32) allowing the DC to investigate with the mentor how to apply the developed methods to some concrete quantum applications.
Joint degree: University of Tilburg, Université Toulouse III-Paul Sabatier.

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