Periodic Reporting for period 2 - TITAN (Tracking Information in Quantum Networks)
Periodo di rendicontazione: 2023-08-01 al 2024-07-31
Recent technological advances allow to build revolutionary devices for information processing by making use of quantum mechanics. Most strikingly, quantum communication allows to transmit information with physical security guarantees and quantum algorithms solve problems that are otherwise computationally unfeasible. Having such quantum technologies in sight, it becomes imperative to investigate what can be achieved in practice but also what the ultimate limitations are. Since both questions are fundamentally concerned with processing information, it is natural to seek an information theoretic approach to find an answer.
Entropy inequalities are among the most important tools in the theory of classical and quantum information processing. They have been successfully applied to bound the information flow in classical information networks and to understand the inner workings of deep neural networks. However, in the quantum case our understanding of such inequalities is currently lacking.
Most importantly, the quantum information community is missing a developed theory regarding conditioning on quantum systems. This fact hinders us in fully judging the potential impact of modern quantum technologies, since one is not able to effectively determine the properties of the underlying operations. I propose to set up a conceptually novel information-theoretic framework to quantify practical advantages of quantum-based devices for network communication and machine learning, by mitigating the quantum conditioning problem. I will develop entropic inequalities that describe the change of entropy under specific operations, in particular, quantum gates and channels that can be part of an encoding or decoding circuit for quantum communication, a quantum internet or act as nodes in a quantum neural network. I will develop a framework that allows to prove such entropy inequalities, on a unified basis and employ them to better understand these important technologies of the future.
Entropy inequalities are among the most important tools in the theory of classical and quantum information processing. They have been successfully applied to bound the information flow in classical information networks and to understand the inner workings of deep neural networks. However, in the quantum case our understanding of such inequalities is currently lacking.
Most importantly, the quantum information community is missing a developed theory regarding conditioning on quantum systems. This fact hinders us in fully judging the potential impact of modern quantum technologies, since one is not able to effectively determine the properties of the underlying operations. I propose to set up a conceptually novel information-theoretic framework to quantify practical advantages of quantum-based devices for network communication and machine learning, by mitigating the quantum conditioning problem. I will develop entropic inequalities that describe the change of entropy under specific operations, in particular, quantum gates and channels that can be part of an encoding or decoding circuit for quantum communication, a quantum internet or act as nodes in a quantum neural network. I will develop a framework that allows to prove such entropy inequalities, on a unified basis and employ them to better understand these important technologies of the future.
During the first two years of the project, I made progress towards several of the tasks laid out in the initial proposal.
In WP1.T1 I proposed to consider information combining via generalized entropies. In the paper "Chain Rules for Renyi Information Combining" with X. Guan and M. Tomamichel we found new relationships between such generalized entropies that allowed us to greatly further our understanding of information combining bounds. In fact, we transferred all known results for the von Neumann setting to this more general quantities. This will allow a new perspective on the remaining problems in future research.
A major focus of the tasks WP1.T2 and WP1.T3 was set on contraction coefficients. Many of the obstacles in the quantum setting stem from the much more complex behavior of the underlying quantities for non-commutative operators. In the recent preprint "Quantum Renyi and f-divergences from integral representations" with M. Tomamichel, we introduce a new family of divergences that have in many regards much more familiar properties than the previously known divergences. This allows us to give several new bounds on entropic quantities and their contraction coefficients in particular. This accomplashes a large part of what we set out to do in WP1.T2 and lays the foundation for further exploration of the resource specific tasks set out in WP1.T3.
As a precursor to the tasks formulated in WP2, I started with the investigation of sequential strategies in several information processing tasks with a focus on hypothesis testing. In the article "Ultimate Limits for Quickest Quantum Change-Point Detection" we investigated the practically relevant setting of quickest change point detection. In "Sequential quantum channel discrimination" we determined optimal asymptotic rates of channel discrimination using these strategies.
In WP1.T1 I proposed to consider information combining via generalized entropies. In the paper "Chain Rules for Renyi Information Combining" with X. Guan and M. Tomamichel we found new relationships between such generalized entropies that allowed us to greatly further our understanding of information combining bounds. In fact, we transferred all known results for the von Neumann setting to this more general quantities. This will allow a new perspective on the remaining problems in future research.
A major focus of the tasks WP1.T2 and WP1.T3 was set on contraction coefficients. Many of the obstacles in the quantum setting stem from the much more complex behavior of the underlying quantities for non-commutative operators. In the recent preprint "Quantum Renyi and f-divergences from integral representations" with M. Tomamichel, we introduce a new family of divergences that have in many regards much more familiar properties than the previously known divergences. This allows us to give several new bounds on entropic quantities and their contraction coefficients in particular. This accomplashes a large part of what we set out to do in WP1.T2 and lays the foundation for further exploration of the resource specific tasks set out in WP1.T3.
As a precursor to the tasks formulated in WP2, I started with the investigation of sequential strategies in several information processing tasks with a focus on hypothesis testing. In the article "Ultimate Limits for Quickest Quantum Change-Point Detection" we investigated the practically relevant setting of quickest change point detection. In "Sequential quantum channel discrimination" we determined optimal asymptotic rates of channel discrimination using these strategies.
Besides the results described in the previous section, I am currently working on several projects to address the remaining tasks left in the proposal.
In particular a manuscript on classification tasks on quantum systems in nearing completion and will address the tasks in WP2.T3.
Furthermore, I am working on extension and applications of the new divergences in our recently published work that should address the remaining questions in WP2.T1 and WP2.T2. Finally, a project addressing dimension bounds for auxiliary quantum random variables is ongoing that would address the remaining questions in WP1.T2.
In particular a manuscript on classification tasks on quantum systems in nearing completion and will address the tasks in WP2.T3.
Furthermore, I am working on extension and applications of the new divergences in our recently published work that should address the remaining questions in WP2.T1 and WP2.T2. Finally, a project addressing dimension bounds for auxiliary quantum random variables is ongoing that would address the remaining questions in WP1.T2.