Periodic Reporting for period 2 - QEntropy (Entropy for Quantum Information Science)
Reporting period: 2024-05-01 to 2025-10-31
Understanding the Limits of Quantum Information: A Mathematical Journey into Quantum Entropy
Summary of the context and overall objectives of the project: Quantum technologies promise to revolutionise how we store, transmit, and process information. At the heart of this revolution lies quantum entropy, and unlike classical entropy which has been mathematically well-understood, quantum entropy remains mathematically elusive. Our project aims to close that gap. By developing new mathematical tools around entropy inequalities, the project seeks to uncover the ultimate physical limits of quantum information processing.
Problem/issue being addressed: In classical systems, entropy is a well-understood concept that underpins modern communication and computation. In the quantum realm, however, the mathematics of entropy becomes far more complex due to the presence of entanglement - strong, non-classical correlations that are a key resource in quantum technologies. While some quantum entropy inequalities, such as the strong sub-additivity inequality, are known and provide insights into entanglement structure, our current general understanding is limited. This restricts our ability to rigorously analyse and exploit quantum systems for tasks like secure communication or efficient computation.
Importance for society: As quantum technologies transition from theory to application, understanding their fundamental limits becomes crucial. Whether designing quantum algorithms for next-generation computers, developing secure quantum communication systems, or simulating new materials for energy and medicine, society increasingly relies on robust mathematical foundations. This project provides exactly that: a deeper understanding of the constraints and capabilities of quantum systems, ultimately guiding the development of practical quantum technologies that are secure, efficient, and reliable.
Overall objectives: The project has two main goals. First, it will develop new mathematical frameworks in matrix analysis and optimisation to describe the behaviour of multiple non-commuting operators. This will lead to the discovery of new entropy inequalities that describe how entanglement is distributed across multipartite quantum systems. Second, the project will translate these mathematical insights into practical tools: approximation algorithms for key problems in quantum information science. This includes advances in quantum cryptography, information theory, and the simulation of strongly correlated quantum systems. Together, these contributions will help define the theoretical boundaries of quantum technologies and bring us closer to realising their full potential.
Summary of the context and overall objectives of the project: Quantum technologies promise to revolutionise how we store, transmit, and process information. At the heart of this revolution lies quantum entropy, and unlike classical entropy which has been mathematically well-understood, quantum entropy remains mathematically elusive. Our project aims to close that gap. By developing new mathematical tools around entropy inequalities, the project seeks to uncover the ultimate physical limits of quantum information processing.
Problem/issue being addressed: In classical systems, entropy is a well-understood concept that underpins modern communication and computation. In the quantum realm, however, the mathematics of entropy becomes far more complex due to the presence of entanglement - strong, non-classical correlations that are a key resource in quantum technologies. While some quantum entropy inequalities, such as the strong sub-additivity inequality, are known and provide insights into entanglement structure, our current general understanding is limited. This restricts our ability to rigorously analyse and exploit quantum systems for tasks like secure communication or efficient computation.
Importance for society: As quantum technologies transition from theory to application, understanding their fundamental limits becomes crucial. Whether designing quantum algorithms for next-generation computers, developing secure quantum communication systems, or simulating new materials for energy and medicine, society increasingly relies on robust mathematical foundations. This project provides exactly that: a deeper understanding of the constraints and capabilities of quantum systems, ultimately guiding the development of practical quantum technologies that are secure, efficient, and reliable.
Overall objectives: The project has two main goals. First, it will develop new mathematical frameworks in matrix analysis and optimisation to describe the behaviour of multiple non-commuting operators. This will lead to the discovery of new entropy inequalities that describe how entanglement is distributed across multipartite quantum systems. Second, the project will translate these mathematical insights into practical tools: approximation algorithms for key problems in quantum information science. This includes advances in quantum cryptography, information theory, and the simulation of strongly correlated quantum systems. Together, these contributions will help define the theoretical boundaries of quantum technologies and bring us closer to realising their full potential.
This project set out to uncover the fundamental mathematical limits of quantum information by deepening our understanding of quantum entropy. Over the course of the reporting period, significant progress was made on multiple fronts, with several key results that push the boundaries of what is known in quantum information theory and its practical applications:
-A Unified Framework for Entanglement Inequalities: We developed a powerful new mathematical framework based on multivariate matrix trace inequalities. This approach unifies most previously known quantum entropy inequalities relevant to entanglement theory. This not only simplifies and streamlines existing proofs but also equips researchers with a flexible and versatile toolbox for identifying new, yet-undiscovered constraints on entanglement in multipartite systems - a critical step in quantifying the advantage of quantum technologies.
-One of the longstanding challenges in network quantum information theory is the so-called quantum joint-typicality problem, which has hindered progress in extending classical network coding theorems to the quantum domain. We introduced a novel mean-zero decomposition lemma that completely sidesteps this obstacle. This result opens the door to translating a wide range of classical information-theoretic protocols to the quantum world, significantly expanding the landscape of feasible quantum communication strategies.
-We studied how reliably quantum channels can be simulated at different rates. Our results revealed that, unlike in classical information theory, there is no critical rate separating different error behaviours. This discovery disproves a widely held conjecture and provides a deeper understanding of the trade-offs between reliability and communication rate in quantum systems. It reshapes expectations for how quantum communication behaves in finite settings, such as real-world systems operating with short code lengths.
-We presented the first quantum algorithm with provable efficiency guarantees for simulating interacting fermionic systems at arbitrary temperature - which can be a classically intractable problem central to materials science and condensed matter physics. Using entropy-based techniques from matrix analysis, we showed that quantum computers can efficiently prepare the Gibbs states of these systems, achieving exponential advantages over any known rigorous classical method. This result demonstrates the real, practical potential of quantum computers to solve problems that are might be out of reach for classical machines.
-We made a major technical contribution by deriving a new class of Bose symmetric quantum de Finetti theorems with linear constraints - something previously thought to be out of reach. By leveraging entropy inequalities that quantify entanglement monogamy, we overcame long-standing technical hurdles. These results are now being applied to improve approximation algorithms for challenging non-convex problems, offering better performance than standard de Finetti approaches and broadening their applicability in quantum optimisation.
Together, these achievements represent a major step forward in our ability to mathematically describe, analyse, and ultimately harness the power of quantum systems for real-world applications in computation, communication, and simulation.
-A Unified Framework for Entanglement Inequalities: We developed a powerful new mathematical framework based on multivariate matrix trace inequalities. This approach unifies most previously known quantum entropy inequalities relevant to entanglement theory. This not only simplifies and streamlines existing proofs but also equips researchers with a flexible and versatile toolbox for identifying new, yet-undiscovered constraints on entanglement in multipartite systems - a critical step in quantifying the advantage of quantum technologies.
-One of the longstanding challenges in network quantum information theory is the so-called quantum joint-typicality problem, which has hindered progress in extending classical network coding theorems to the quantum domain. We introduced a novel mean-zero decomposition lemma that completely sidesteps this obstacle. This result opens the door to translating a wide range of classical information-theoretic protocols to the quantum world, significantly expanding the landscape of feasible quantum communication strategies.
-We studied how reliably quantum channels can be simulated at different rates. Our results revealed that, unlike in classical information theory, there is no critical rate separating different error behaviours. This discovery disproves a widely held conjecture and provides a deeper understanding of the trade-offs between reliability and communication rate in quantum systems. It reshapes expectations for how quantum communication behaves in finite settings, such as real-world systems operating with short code lengths.
-We presented the first quantum algorithm with provable efficiency guarantees for simulating interacting fermionic systems at arbitrary temperature - which can be a classically intractable problem central to materials science and condensed matter physics. Using entropy-based techniques from matrix analysis, we showed that quantum computers can efficiently prepare the Gibbs states of these systems, achieving exponential advantages over any known rigorous classical method. This result demonstrates the real, practical potential of quantum computers to solve problems that are might be out of reach for classical machines.
-We made a major technical contribution by deriving a new class of Bose symmetric quantum de Finetti theorems with linear constraints - something previously thought to be out of reach. By leveraging entropy inequalities that quantify entanglement monogamy, we overcame long-standing technical hurdles. These results are now being applied to improve approximation algorithms for challenging non-convex problems, offering better performance than standard de Finetti approaches and broadening their applicability in quantum optimisation.
Together, these achievements represent a major step forward in our ability to mathematically describe, analyse, and ultimately harness the power of quantum systems for real-world applications in computation, communication, and simulation.
This project has achieved several results that go well beyond the original goals, challenging previous assumptions, and opening applications in quantum information science.
Quantum Algorithms Outperform Classical Methods in Simulating Strongly Correlated Matter: One of the most surprising developments came from our work on quantum Gibbs samplers. Originally, the project aimed to develop classical approximation algorithms for simulating quantum many-body systems - a notoriously hard problem. Instead, we discovered a provably efficient quantum algorithm capable of preparing thermal states for complex, interacting systems like the Fermi-Hubbard model. This is a key model in condensed matter physics and materials science, and simulating it accurately has long been out of reach for classical computers. Our results show that quantum computers can provide an exponential advantage for problems of direct physical relevance, pushing quantum simulation from a theoretical promise to a practical tool.
Bypassing an Old Bottleneck in Quantum Shannon Theory: The so-called quantum joint typicality problem has stood as a major obstacle in extending classical communication theory to the quantum domain. While we did not solve this problem directly, we instead showed that it can be effectively circumvented in many cases. Using a new multipartite mean-zero decomposition lemma, we found a conceptually simple yet powerful workaround that allows classical coding theorems to be lifted into the quantum setting. This insight not only simplifies the theory but has already begun inspiring new research, with several follow-up works by the community appearing soon after our publication.
Overcoming Barriers in Quantum Optimisation: We addressed a long-standing open question: can one formulate Bose-symmetric quantum de Finetti theorems with linear constraints? These tools are central to understanding symmetric quantum systems and had been thought potentially unachievable under such constraints. Surprisingly, by applying entropy inequalities related to the monogamy of entanglement, we were able to derive elegant new versions of these theorems. This result not only advances the theory of quantum correlations but also significantly improves the practical efficiency of numerical algorithms used to approximate solutions in non-commutative polynomial optimisation - relevant for a wide range of applications in quantum information theory.
These advances underscore the profound impact that mathematical insight can have on quantum information theory, yielding not only deeper theoretical understanding but also powerful, practical tools that are already shaping the field. Building on this momentum, we will continue to pursue all four original research directions: (i) developing mathematical techniques for quantum entropy inequalities, (ii) applying these tools to fundamental questions in quantum information theory, (iii) advancing the theory and practice of quantum cryptography, and (iv) efficiently resolving phenomena in quantum many-body physics. The overarching goal remains to understand the structure of multipartite entanglement spectra, paving the way for new approximation algorithms across quantum information science.
Quantum Algorithms Outperform Classical Methods in Simulating Strongly Correlated Matter: One of the most surprising developments came from our work on quantum Gibbs samplers. Originally, the project aimed to develop classical approximation algorithms for simulating quantum many-body systems - a notoriously hard problem. Instead, we discovered a provably efficient quantum algorithm capable of preparing thermal states for complex, interacting systems like the Fermi-Hubbard model. This is a key model in condensed matter physics and materials science, and simulating it accurately has long been out of reach for classical computers. Our results show that quantum computers can provide an exponential advantage for problems of direct physical relevance, pushing quantum simulation from a theoretical promise to a practical tool.
Bypassing an Old Bottleneck in Quantum Shannon Theory: The so-called quantum joint typicality problem has stood as a major obstacle in extending classical communication theory to the quantum domain. While we did not solve this problem directly, we instead showed that it can be effectively circumvented in many cases. Using a new multipartite mean-zero decomposition lemma, we found a conceptually simple yet powerful workaround that allows classical coding theorems to be lifted into the quantum setting. This insight not only simplifies the theory but has already begun inspiring new research, with several follow-up works by the community appearing soon after our publication.
Overcoming Barriers in Quantum Optimisation: We addressed a long-standing open question: can one formulate Bose-symmetric quantum de Finetti theorems with linear constraints? These tools are central to understanding symmetric quantum systems and had been thought potentially unachievable under such constraints. Surprisingly, by applying entropy inequalities related to the monogamy of entanglement, we were able to derive elegant new versions of these theorems. This result not only advances the theory of quantum correlations but also significantly improves the practical efficiency of numerical algorithms used to approximate solutions in non-commutative polynomial optimisation - relevant for a wide range of applications in quantum information theory.
These advances underscore the profound impact that mathematical insight can have on quantum information theory, yielding not only deeper theoretical understanding but also powerful, practical tools that are already shaping the field. Building on this momentum, we will continue to pursue all four original research directions: (i) developing mathematical techniques for quantum entropy inequalities, (ii) applying these tools to fundamental questions in quantum information theory, (iii) advancing the theory and practice of quantum cryptography, and (iv) efficiently resolving phenomena in quantum many-body physics. The overarching goal remains to understand the structure of multipartite entanglement spectra, paving the way for new approximation algorithms across quantum information science.