Quantum Nonlocality in Bilocal Networks: An Operator Algebraic Perspective

In this thesis, we explore the applications of the theory of operator algebras, such as C ∗ -algebras and von Neumann algebras, in quantum physics with a focus on the quantum bilocal network scenario. Originating from Heisenberg’s picture, the theory of operator algebras has become a fruitful field and a powerful mathematical tool in the study of quantum systems. Quantum nonlocality is essential for understanding the foundations of quantum mechanics, and quantum correlations in network scenarios provide a robust framework for its investigation. We combine the algebraic perspective with the study of quantum nonlocality to analyse the quantum bilocal network scenario, involving three parties, Alice, Bob, and Charlie, arranged linearly. In this scenario, two independent quantum sources are present, one shared between Alice and Bob, and the other shared between Bob and Charlie. Recent developments from [57, Renou & Xu, arXiv:2210.09065] and [35, Ligthart & Gross, arXiv:2212.11299] demonstrate the applicability of operator algebras in this context, offering a deeper understanding of quantum nonlocality in network settings. This thesis serves as a reference for results in the theory of operator algebras relevant to the study of quantum nonlocality, introducing notable results and attempting to draw connections between them. By building upon the findings of the two papers mentioned above, the thesis highlights the potential of these abstract mathematical tools in uncovering new insights into quantum mechanics and motivates further research in the field using an operator algebraic perspective.

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