# Workshop GPT&QIT
Generalized Probabilistic Theories and Quantum Information

### Lyon, July 1st-2nd, 2019

### Mini-course

** Ludovico Lami (Nottingham) : an introduction to General Probabilistic Theories **

**Abstract**: This series of lectures aims to provide an introduction to general probabilistic theories (GPTs), with focus on the physical motivations behind it, the mathematical aspects of the
formalism, and some
of the most pressing outstanding problems. The mini-course is composed by four lectures of 1 1/2 hours each, structured as follows:
- (1) We will review the basic formalism of GPTs in finite
dimension,
discussing the main assumptions and some instructive examples. To this end, the basic theory of ordered vector spaces is also introduced.
- (2) Moving on to bipartite systems, we will see how a tensor
product structure emerges from physically reasonable axioms. The set of allowed bipartite states is not uniquely determined, yet it is subjected to some nontrivial constraints. Minimal and maximal
tensor products of cones are introduced, together with the concepts of abstract entanglement and nonlocality. Depending on time, we may look into some applications to data hiding.
- (3) The third
lecture is devoted to the notion of classical system. We will prove some characterisation results and use them to give some partial answers to a pressing open question: what pairs of systems are
unentangleable?
- (4)
We will go back and examine the foundations of GPTs. After some necessary preliminaries on the theory of ordered topological vector spaces, we will state an important theorem
proved by Günther Ludwig at the end of the 1960s, which derives the GPT formalism from few physically motivated axioms. Some proofs ideas are discussed.

**Research talks**

** Andreas Bluhm (Munich) : A strengthened data processing inequality for the Belavkin-Staszewski relative entropy **

**Abstract**: see arXiv preprint

** Anna Jenčová (Bratislava) : Incompatible measurements in GPT **

**Abstract**: General probabilistic theories provide a framework for studying incompatibility of measurements from a geometric point of view. In this talk, we concentrate on collections of
two-outcome measurements, or effects. We discuss characterizations of (in)compatible collections of measurements and introduce the notion of an incompatibility witness. We study some notions of
incompatibility degree and show a geometric characterization of the maximal incompatibility degree of a collection of effects, attainable in a given theory.

** Maria Jivulescu (Timisoara) : Random positive operator valued measures**

**Abstract**: We introduce several notions of random positive operator valued measures
(POVMs), and we prove that some of them are equivalent. We then study
statistical properties of the effect operators for the canonical examples,
obtaining limiting eigenvalue distributions with the help of free
probability theory. Similarly, we obtain the large system limit for several
quantities of interest in quantum information theory, such as the sharpness,
the noise content, and the probability range. Finally, we study different
compatibility criteria, and we compare them for generic POVMs.
Joint work with Teiko Heinosaari and I. Nechita, see arXiv preprint.

** Alexander Müller-Hermes (Copenhagen) : Positivity of linear maps under tensor powers **

**Abstract**: Both completely positive and completely copositive maps stay positive under tensor powers, i.e., under tensoring the linear map with itself. But are there other examples of maps
with this
property? We discuss the connection of this question to the NPPT bound entanglement problem, and give a negative answer for maps on 2-dimensional systems. We will then consider more general
questions for other classes of linear maps. In particular we will show that any decomposable map, that is neither completely positive nor completely copositive, will lose decomposability eventually
after taking enough tensor powers. As an application this gives rise to new examples of non-decomposable positive maps.

** Martin Plávala (Bratislava) : No-free-information principle in general probabilistic theories **

**Abstract**: In quantum theory, the no-information-without-disturbance and no-free-information principles express that those observables that do not disturb the measurement of another observable
and those that can
be measured jointly with any other observable must be trivial, i.e., coin tossing observables. We show that in the framework of general probabilistic theories these statements do not hold in general
and continue to completely specify these two classes of observables. In this way, we obtain characterizations of the probabilistic theories where these statements hold. See
arXiv preprint
### Location

All talks take place in the Braconnier building, room 112 (1st floor). The building is just in front of the tramway stop "Université Lyon 1".
### Schedule

Here is the schedule.
### Participants

Simon Andreys (Lyon)
Guillaume Aubrun (Lyon)
Ivan Bardet (Cambridge)
Denis Bernard (Paris)
Andreas Bluhm (Munich)
Omar Fawzi (Lyon)
Anna Jencova (Bratislava)
Maria Jivulescu (Timisoara)
Ludovico Lami (Nottinhgam)
Alexander Müller-Hermes (Copenhagen)
Ion Nechita (Toulouse)
Martin Plavala (Bratislava)
Christoper Salinas Zavala (Saint-Étienne)
Jean-Marie Stéphan (Lyon)
Shang-Chun Yu (Lyon)
If you would like to participate, please tell me: aubrun[AT]math.univ-lyon1.fr.

**Funding**

The workshop is funded by the ANR Grant Stoq