Periodic Reporting for period 1 - PENNSION (Partition and accumulation of ENtropy in infinite-dimeNSIONs)
Berichtszeitraum: 2023-08-01 bis 2025-07-31
Quantum information theory lies at the heart of the next technological revolution, offering transformative possibilities in computation, communication, and cryptography. As Europe advances its strategic agenda through initiatives like the Quantum Flagship, there is a growing need to establish a rigorous theoretical foundation for quantum technologies that not only captures the operational aspects but also aligns with the physical realities of quantum systems, many of which are inherently infinite-dimensional. This project addressed a key gap in the existing theoretical landscape: the lack of a comprehensive understanding of foundational theorems such as the Asymptotic Equipartition Property (AEP) and the Entropy Accumulation Theorem (EAT) in the general setting of von Neumann algebras, which provide the mathematical language for infinite-dimensional quantum systems. These theorems are central to quantum information processing tasks, including data compression, statistical inference, and security proofs in cryptography.
The primary objective of the project was to extend these cornerstone results to the infinite-dimensional setting and thereby offer a rigorous and broadly applicable framework for quantum information theory beyond the limitations of finite-dimensional models. Achieving this required a deep synthesis of ideas from operator algebras, quantum physics, and information theory — reflecting a truly interdisciplinary effort that draws from both the physical sciences and mathematical foundations. The project also explored how these results apply under structural constraints, such as those arising from subalgebras, which naturally appear in many real-world scenarios involving symmetries, partial observations, or operational limitations.
The anticipated impact of this work is both theoretical and strategic. By strengthening the mathematical infrastructure underpinning quantum technologies. It opens up new avenues for applying advanced mathematical tools to practical quantum protocols, including those used in secure communication and quantum cryptography. At scale, these results contribute to Europe's leadership in quantum science, helping ensure that the foundational understanding of quantum systems keeps pace with technological innovation.
The primary objective of the project was to extend these cornerstone results to the infinite-dimensional setting and thereby offer a rigorous and broadly applicable framework for quantum information theory beyond the limitations of finite-dimensional models. Achieving this required a deep synthesis of ideas from operator algebras, quantum physics, and information theory — reflecting a truly interdisciplinary effort that draws from both the physical sciences and mathematical foundations. The project also explored how these results apply under structural constraints, such as those arising from subalgebras, which naturally appear in many real-world scenarios involving symmetries, partial observations, or operational limitations.
The anticipated impact of this work is both theoretical and strategic. By strengthening the mathematical infrastructure underpinning quantum technologies. It opens up new avenues for applying advanced mathematical tools to practical quantum protocols, including those used in secure communication and quantum cryptography. At scale, these results contribute to Europe's leadership in quantum science, helping ensure that the foundational understanding of quantum systems keeps pace with technological innovation.
This project focused on extending two foundational theorems in quantum information theory — the Asymptotic Equipartition Property (AEP) and the Entropy Accumulation Theorem (EAT) — to the setting of infinite-dimensional quantum systems described by von Neumann algebras. To achieve this, I developed a range of advanced technical tools at the intersection of operator algebras, quantum probability, and information theory. These included entropy inequality techniques, martingale convergence and Haagerup approximation in general von Neumann algebras, and operator algebraic approximations of typical subspaces. The work involved close collaboration with leading experts in the field, enabling the exchange of ideas across disciplines and contributing to the refinement and validation of the developed methods. One of the key scientific outcomes was a rigorous formulation and proof of the generalized AEP result for states and quantum channels in this infinite-dimensional framework, providing a solid basis for asymptotic analysis in a much broader class of quantum systems. On the same account a chain rule for sandwiched Rényi entropies has been obtained in a von Neumann algebra context.
Building on this, I also made significant progress toward a general formulation of the Entropy Accumulation Theorem in the (approximately finite dimensional) von Neumann algebraic setting, a direction with strong implications for quantum cryptography and statistical mechanics. These achievements mark an important step in bridging the gap between abstract mathematical theory and the operational needs of quantum information science. Alongside these core research goals, the project contributed to the development of a broader international research network focused on operator algebraic methods in quantum theory, creating new opportunities for collaboration and dissemination of ideas.
Building on this, I also made significant progress toward a general formulation of the Entropy Accumulation Theorem in the (approximately finite dimensional) von Neumann algebraic setting, a direction with strong implications for quantum cryptography and statistical mechanics. These achievements mark an important step in bridging the gap between abstract mathematical theory and the operational needs of quantum information science. Alongside these core research goals, the project contributed to the development of a broader international research network focused on operator algebraic methods in quantum theory, creating new opportunities for collaboration and dissemination of ideas.
The project has delivered several results that significantly advance the state of the art in quantum information theory, particularly in extending key operational principles to infinite-dimensional systems. By rigorously formulating and proving the Asymptotic Equipartition Property (AEP) and laying the groundwork for the Entropy Accumulation Theorem (EAT) in the general framework of von Neumann algebras, this research has overcome a major theoretical bottleneck that previously limited the applicability of these results to finite-dimensional quantum systems. These developments now make it possible to analyze quantum information tasks — such as hypothesis testing, randomness extraction, and quantum key distribution — in more realistic models where infinite-dimensional structures naturally arise, such as in quantum optics, continuous-variable systems, and quantum field theory.
The project not only introduced new mathematical tools and techniques to the field but also established a methodological bridge between abstract operator algebra theory and the operational needs of quantum information science. This opens up new directions for both theoretical inquiry and application-driven development. For further uptake and success, continued interdisciplinary collaboration between mathematicians, computer scientists, and physicists will be essential, particularly in translating these foundational results into practical protocols and security proofs. There is also scope for integrating these findings into broader standardization and certification frameworks in quantum technologies, especially in areas like quantum cryptography and secure communications. Due to an early transition into a faculty position, this Marie Curie project was concluded five months ahead of schedule, which resulted in a few of the originally planned objectives remaining unrealised, specially the investigation into quantum cryptography. This part will be carried out in the future endeavour. As quantum technologies continue to mature and attract industrial interest, further research investment, support for collaborative networks, and engagement with emerging regulatory and standardization bodies will be key to ensuring the long-term impact and utility of the results generated by this project.
The project not only introduced new mathematical tools and techniques to the field but also established a methodological bridge between abstract operator algebra theory and the operational needs of quantum information science. This opens up new directions for both theoretical inquiry and application-driven development. For further uptake and success, continued interdisciplinary collaboration between mathematicians, computer scientists, and physicists will be essential, particularly in translating these foundational results into practical protocols and security proofs. There is also scope for integrating these findings into broader standardization and certification frameworks in quantum technologies, especially in areas like quantum cryptography and secure communications. Due to an early transition into a faculty position, this Marie Curie project was concluded five months ahead of schedule, which resulted in a few of the originally planned objectives remaining unrealised, specially the investigation into quantum cryptography. This part will be carried out in the future endeavour. As quantum technologies continue to mature and attract industrial interest, further research investment, support for collaborative networks, and engagement with emerging regulatory and standardization bodies will be key to ensuring the long-term impact and utility of the results generated by this project.