My research interests lie in the intersection of information theory, cryptography and quantum mechanics. My main focus is on the mathematical foundations of quantum information theory, for example the study of entropy and other information measures, as well as theoretical questions that arise in quantum communication and cryptography when the available resources are limited.
Information processing with finite resources
One of the predominant challenges when engineering future quantum information processors is that complex quantum systems are notoriously hard to prepare and control coherently. Consequently, there will be limitations on the size of quantum computers for the foreseeable future. For this reason I investigate quantum information processing under the assumption that the available physical resources are limited. For example, we want to understand the fundamental limits that restrict information transmission between two parties that possess a small quantum device:
- M. Tomamichel, M. Berta, and J. M. Renes, “Quantum coding with finite resources”, Nature Communications 7, 11419 (2016);
- M. Tomamichel and V. Y. F. Tan, “Second-order asymptotics for the classical capacity of image-additive quantum channels”, Communications in Mathematical Physics 338 (1), 103–137 (2015).
My blog post on the first of these papers describes this research in more detail.
Cryptography in a quantum world
Securing information against potential adversaries is a ubiquitous challenge in our modern world, appearing in diverse guises such as sending private email and online commerce, among myriad others. Quantum cryptography studies secure information processing using quantum devices. I am interested in security proofs for various cryptographic schemes—mathematical arguments that certify that the behavior of a given protocol is indeed secure. For example, we show that quantum key distribution allows two parties to efficiently produce a shared key that is secret from any eavesdropper:
- M. Tomamichel, C. C. W. Lim, N. Gisin, and R. Renner, “Tight finite-key analysis for quantum cryptography”, Nature Communications 3, 634 (2012).
Recently we have analyzed vulnerabilities of cryptocurrencies like bitcoin to attacks by quantum computers, finding that potentially devastating attacks might be possible in as early as ten years. We also suggested effective countermeasures. Our preprint can be found on [arXiv], and has garnered significant media attention.
Mathematical foundations of quantum information theory
In order to solve questions that arise in quantum information theory we often require new technical ingredients and thus it is necessary to continuously expand the available mathematical toolkit. In particular, I am interested in exploring various measures of entropy, information and correlation, such as Rényi entropies. For example, in the following paper we introduce a new quantum generalization of these measures that has already found many applications:
- M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel, “On quantum Rényi entropies: a new generalization and some properties”, Journal of Mathematical Physics 54 (12), 122203 (2013).
More generally, new results on mathematical properties of information and correlation measures often have various applications to information theory beyond the specific applications that originally inspired the research. As such, progress in information theory often goes hand in hand with a more thorough understanding of the mathematical framework underlying it.