Over the past century, quantum theory has had profound impact on the development of pure and applied mathematics. The mathematical underpinnings of quantum mechanics led to the theory of operator algebras, and quantum groups were invented to study quantum integrable systems. Later on, quantum information theory emerged as framework for processing information which makes use of the rich features of quantum theory.
This programme will focus on spectacular new connections between these fields. Over the past decade, there has been an explosion of interest in understanding probability distributions arising from models of entangled quantum systems. Quantum groups enter the picture by capturing the natural notion of symmetry in this context. At the same time, extensive links have been established with long-standing problems in the area of operator algebras. A ground-breaking result is the proposed resolution of the Connes Embedding Problem by Ji, Natarajan, Vidick, Wright and Yuen from 2020. This uses methods from quantum information theory to attack a fundamental problem about the approximation of operator algebras by finite building blocks.
These developments have opened up completely new research directions, which this programme aims to explore.
Over the past decade, there has been an explosion of interest in the theory of quantum correlations. Over the same time period, extensive links have been established with long-standing problems in operator algebras. Recent ground-breaking results in this area are the proposed resolution of the Connes Embedding problem by Ji, Natarajan, Vidick, Wright and Yuen using methods from quantum complexity theory, and the work of Slofstra on separating different classes of quantum correlations. Simultaneously, fascinating links have emerged between quantum information theory and the theory of compact quantum groups.
The intersection between graph theory, quantum information and compact quantum groups offers a wealth of open problems and questions, which will be one of the focus points of the programme. The recent notion of a quantum graph should play an important role in this context. Quantum graphs include the confusability graphs of quantum channels and arise naturally in the study of quantum automorphism groups. At the same time the first random constructions of quantum graphs have started to appear, offering exciting perspectives in view of the large impact and importance of probabilistic methods on classical graph theory.
The impact of techniques from quantum information theory and quantum complexity theory in pure mathematics, and particularly operator algebras, is only at its beginning. The negative resolution of Connes' Embedding problem poses a number of challenges, such as the question of the existence of non-hyperlinear groups. There also remains much work to fully understand the work by Ji, Natarajan, Vidick, Wright and Yuen and its ramifications, and the programme aims to contribute to this effort.
The programme will bring together leading experts and young researchers from quantum information, quantum groups, and operator algebras. Due to the discovery of the completely unexpected links outlined above there is significant potential for further interaction between these communities. A key prerequisite, however, for future success in exploring the opportunities emerging from these developments is a fruitful exchange of technology and ideas. Currently, researchers in these fields often use different terminology, which is a barrier to easy collaboration. One of the key aims of our programme is to reduce this barrier by helping researchers with different scientific backgrounds to start to speak a common language.
The Institute kindly requests that any papers published as a result of this programme’s activities are credited as such. Please acknowledge the support of the Institute in your paper using the following text:
The author(s) would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Quantum information, quantum groups and operator algebras, where work on this paper was undertaken. This work was supported by EPSRC grant EP/R014604/1.
4 November 2024 to 8 November 2024
2 December 2024 to 6 December 2024
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