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Eigenvalues of tensors – applications to multipartite entanglement



Lab: LPT

Duration: 5 months full-time internship
6 months full-time internship

Latest starting date: 01/06/2024

Localisation: LPT Toulouse


This research master's degree project could be followed by a PhD

Work package:
Quantum entanglement is one of the most intriguing phenomena in quantum physics, and it plays a crucial role in various quantum technologies, including quantum computing and quantum communication. Multipartite quantum entanglement, which involves the entanglement of more than two quantum systems, is an area of increasing interest due to its potential applications in quantum information processing. In recent years, tensor networks have emerged as a powerful tool for understanding and characterizing multipartite entanglement in quantum systems. This proposal outlines a research project that aims to explore the fascinating intersection of multipartite quantum entanglement and the mathematics of tensors. We seek to investigate the theoretical foundations, develop computational tools, and explore practical applications of these topics. Objectives - Theoretical Investigation: We will delve into the mathematical and theoretical aspects of multipartite quantum entanglement and tensors (norms, eigenvalues, etc). This includes studying the principles of quantum entanglement in multipartite systems, tensor representations of quantum states, and the concepts of tensor norms and eigenvalues. Development of Computational Tools: We will develop computational algorithms and software tools for efficiently describing and simulating multipartite entangled quantum states. Characterization and Classification: Our project will focus on characterizing different types of multipartite entanglement in quantum systems. We will aim to classify multipartite entangled states and investigate their properties using the mathematical tools developed.

- Guillaume Aubrun and Stanislaw J. Szarek. Alice and Bob Meet Banach: The Interface of Asymptotic Geometric Analysis and Quantum Information Theory, volume 223. American Mathematical Soc., 2017. - Shmuel Friedland and Lek-Heng Lim. Nuclear norm of higher-order tensors. Mathematics of Computation, 87(311):1255– 1281, 2018. - Alexandre Grothendieck. Résumé de la théorie métrique des produits tensoriels topologiques. Soc. de Matematica de Sao Paulo, 1956. - Christopher J Hillar and Lek-Heng Lim. Most tensor problems are np-hard. Journal of the ACM (JACM), 60(6):45, 2013. Maria Anastasia Jivulescu, C ́ecilia Lancien, and Ion Nechita. - Multipartite entanglement detection via projective tensor norms. arXiv preprint arXiv:2010.06365, 2020. - Tao Li, Le-Min Lai, Deng-Feng Liang, Shao-Ming Fei, and Zhi-Xi Wang. Entanglement witnesses based on symmetric informationally complete measurements. International Journal of Theoretical Physics, pages 1–9, 2020. - Oliver Rudolph. A separability criterion for density operators. Journal of Physics A: Mathematical and General, 33(21):3951, 2000. - Tzu-Chieh Wei and Paul M Goldbart. Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Physical Review A, 68(4):042307, 2003.

Areas of expertise:
quantum information theory, quantum computing, quantum entanglement, linear algebra, tensors

Required skills for the internship:
solid background in quantum theory and linear algebra