### Quantum Shannon Theory & Quantum Cryptography

My research is in quantum information theory, with a focus on quantum Shannon theory and quantum cryptography. The former is about the general theory of information processing in the quantum setting, whereas the latter is more specifically about techniques for secure communication in the presence of malicious parties. Good examples for my work are the following results:

**The uncertainty principle in the presence of quantum memory**

*Mario Berta, Matthias Christandl, Roger Colbeck, Joseph M. Renes, and Renato Renner*

Manifold applications of this extended uncertainty principle – in particular in quantum cryptography – are discussed in our recent review article Entropic uncertainty relations and their applications.

**One-shot decoupling**

*Frédéric Dupuis, Mario Berta, Jürg Wullschleger, and Renato Renner*

Applications of decoupling range from quantum Shannon theory to quantum cryptography to quantum thermodynamics to quantum many body physics to the study of black hole radiation. Recently, we were able to extend our results to Catalytic decoupling of quantum information and Conditional decoupling of quantum Information.

### Matrix Analysis & Optimization Theory

In my work I focus on the underlying mathematical methods, including in particular matrix analysis and (non-commutative) optimization theory. Good examples for this fruitful interplay are the following results:

**Multivariate trace inequalities**

*David Sutter, Mario Berta, and Marco Tomamichel*

From the abstract: We prove several trace inequalities that extend the Golden-Thompson and the Araki-Lieb-Thirring inequality to arbitrarily many matrices. Our proofs rely on complex interpolation theory as well as asymptotic spectral pinching, providing a transparent approach to treat generic multivariate trace inequalities. As an application, we prove explicit remainder terms for the monotonicity of the quantum relative entropy and strong sub-additivity of the von Neumann entropy in terms of recoverability, which are tight in the commutative case.

**Quantum bilinear optimization**

*Mario Berta, Omar Fawzi and Volkher B. Scholz*

From the abstract: We study optimization programs given by a bilinear form over non-commutative variables subject to linear inequalities. We introduce an asymptotically converging hierarchy of efficiently computable semidefinite programming (SDP) relaxations for this quantum optimization. As applications we study the entangled value of two-prover games, entanglement assisted channel coding, quantum-proof randomness extractors, and the positive semidefinite cone as introduced by Laurent and Piovesan.

Some introductory books and notes can be found under links. (More detailed research statement available upon request.)