## Mathematics of Quantum Information

###### On **Friday 4th May **the CCIMI will host a one day conference on the** Mathematics of Quantum Information**.

###### The event will be held in MR2 at the **Centre for Mathematical Sciences**, Cambridge University, and will feature a number of international experts on Quantum Information. Talks are aimed to be accessible to a mathematical, but non-specialised audience.

Quantum information science is the study of information encoded in quantum mechanical systems, including the development of algorithms designed for quantum computers, the study of the processing and transmission of quantum information, and the realizability of information-theoretic tasks. This theory has important differences from classical information theory due to exclusively quantum mechanical phenomena such as entanglement, and understanding these differences is one of the goals of the field. In particular, understanding entanglement itself, as well as how it can be transformed and used is an active area of research. Moreover, while information science often considers information in the abstract, one can also answer information-theoretic questions about physical systems of interest.

Quantum information science is a broad and active field; in this workshop, Fernando Brandao will discuss quantum algorithms for semidefinite programming; Omar Fawzi will discuss the relationship between quantum conditional mutual information and the operational task of local recovery; Barbara Kraus will discuss multipartite entanglement; Yan Pautrat will discuss Landauer’s Principle, relating energies and entropies in repeated interaction systems, and David Pérez Garćia will discuss the computational decidability of problems in quantum many body systems.

###### Registration is free and includes lunch and refreshments, please **sign up **at https://www.eventbrite.co.uk/e/mathematics-of-quantum-information-tickets-45060196260. **Registration closes at midday on Tuesday May 1st.**

**Programme:**

**10:15 – 10:50 ** **Coffee and Registration**

**10:50 – 11:00 Welcome from Carola **

**11:00 – 12:00 Fernando Br****andao (** – New Directions in Quantum Algorithms: Thermalization meets Convex Optimization

Abstract: Quantum computers hold the promise of solving certain problems much faster than classical devices. An important challenge in quantum computing is to come up with more quantum algorithms offering speed-ups. I will discuss recent results on quantum algorithms for semidefinite programming, an important class of convex optimization problems with widespread applications (from resource allocation to approximating hard combinatorial problems). I will show how solving semidefinite programs (SDPs) is connected to the task of quantum Gibbs sampling (which consists of computing properties of thermal states at finite temperature on a quantum computer). I will then discuss results on the time of thermalization of many-body quantum systems and show that they directly give quantum speed-ups for SDPs. I will also argue that the quantum algorithm for SDPs can be seen as a generalization of quantum annealing and is a good candidate for realisation on small quantum computers.

**12:00 – 13:00 Omar Fawzi (** – Quantum conditional mutual information

Abstract: The Shannon and von Neumann entropies quantify the uncertainty in a system. They are operationally motivated by natural information processing tasks such as compression or randomness extraction. In addition to characterizing the fundamental rates at which such tasks can be performed, their additivity properties make them a very valuable tool in applications ranging from complexity theory to many-body systems.

A difficulty that arises when dealing with quantum systems is the operational meaning of quantum observers, or the way to interpret conditional entropies. A particularly interesting quantity is the mutual information between two systems conditioned on a third quantum system. What notion of conditional independence does this quantity measure? After an overview of some operational interpretations of conditional entropies, I will show how the quantum conditional mutual information can be related to the task of local recovery.

**13:00 – 14:00 Lunch**

**14:00 – 15:00 David Pérez García (** – Undecidability in Quantum Physics: Hilbert’s Second and Sixth Problems Meet

Abstract: The pioneering work of Goedel and Turing in the 30s showed that there exist problems in mathematics and computer science that cannot be solved. They are called undecidable. Since then, several problems in physics have been shown to be undecidable too. In this talk I will show that many interesting properties of a quantum many body system are indeed undecidable. This negative result has, however, a positive side. It predicts the existence of a new effect that we name “size-driven quantum phase transition”. I will present this effect, its characteristic features, as well as our recent ideas to try to observe it.

**15:00 – 15:30 Coffee**

**15:30 – 16:30 Yan Pautrat (
**Abstract: We study Landauer’s principle for repeated interaction systems consisting of a reference quantum system S in contact with an environment E made of a chain of independent quantum probes. The system S interacts with each probe sequentially, and the Landauer principle relates the energy variation of E and the decrease of entropy of S. We consider the adiabatic regime where the chain contains T probes and displays variations of order 1/T between the successive probes. We consider refinements of the Landauer bound at the level of the full statistics (FS) associated with a two-time measurement protocol. At the technical level, our results rely on a non-unitary adiabatic theorem and and an analysis of the spectrum of complex deformations of families of irreducible completely positive trace-preserving maps. This is joint work with Eric Hanson, Alain Joye and Renaud Raquépas. – Landauer’s Principle in Repeated Interaction Systems

**16:30 – 17:30 Barbara Kraus (**– Multipartite SLOCC and LOCC transformations

Abstract: In this talk I will focus on the local manipulation of multipartite entanglement contained in systems which are composed of n d-level subsystems. I will explain that non-trivial LOCC (Local operations assisted by classical communication) transformations among generic fully entangled pure states are almost never possible. Hence, almost all multipartite states are isolated. They can neither be deterministically obtained from local unitary (LU)-inequivalent states via local operations, nor can they be deterministically transformed to pure fully entangled LU-inequivalent states. I will then present a simple and elegant expression for the maximal probability to convert one multi-qudit fully entangled state to another for this generic set of states. The consequences of these findings in the context of entanglement theory will be discussed. Moreover, I will present some recent results about probabilistic transformations from a higher dimensional Hilbert space to a lower dimensional one.

**17:30 – 18:30 Drinks Reception**