Interactions between operator space theory and quantum probability with applications to quantum information – OSQPI

The interplay between operator spaces and quantum probability has started to emerge in the last years and shown to be more than fruitful. But applications of these theories to quantum information began to appear only very recently. Noncommutative Lp-spaces are at the intersection of these areas and play a crucial role in recent mathematical research motivated by concepts and problems from quantum physics. Even though it is not new, the noncommutative integration has recently regained considerable interest among researchers from different fields such as functional analysis, mathematical physics, quantum probability and quantum information. These “quantum mathematics” are at the frontier between theoretical physics and mathematics. Their development will certainly have a profound impact to implementing quantum algorithms and information theory. The first interactions between operator spaces, quantum probability and quantum information are already impressive, one can certainly expect that their full exploration will open a new vast avenue of perspectives and impact the future development of these areas.

Some key results in the last ten years were obtained through sporadic explorations of interactions between these quantised theories. It would be appropriate to fully investigate and develop these links, in particular through the applications of noncommutative Lp-spaces, noncommutative martingale and matricial inequalities. These inequalities of quantum probabilistic nature have natural applications to theoretical and mathematical physics and quantum information. The originality of this project lies in the systematic exploration of these interactions to build up a coherent program that goes far beyond the current state of the art. We will exploit particularly our original and novel approaches to these fields where operator spaces or noncommutative Lp-spaces have been rarely or have not been explored so far. We strongly believe that the research in quantum probability and quantum information is at a crossroad, and that this is the right time to investigate them through the insight of operator spaces and noncommutative Lp-spaces. This project will initiate new studies across a wide spectrum of fields at the frontiers in noncommutative analysis.

This project will follow three directions. The first one is devoted to the study of operator spaces by using quantum probabilistic models. It deals with Grothendieck's program which requires notably the concrete realisation of operator spaces. Another aspect concerns the study of fundamental examples of completely bounded maps such as Fourier or Schur multipliers and their applications to approximation properties. The second direction goes conversely and treats the exploration of quantum probability by operator spaces. The principal and ambitious objective here is the creation of an analytic theory of quantum stochastic integration which now seems possible thanks to the noncommutative martingale inequalities. This analytic theory of quantum stochastic integration should open a large avenue of applications exactly like Itô's integral in the classical probability theory. Markov dilations and deformations of quantum groups or Fock spaces will be part of our objectives in this second direction too. The third one deals with applications of operator spaces and quantum probability to quantum information. We intend, in particular to study various Bell type inequalities and quantum entanglement. The different notions of entropy or capacity and their additivity problem will be another important aspect of this part. Finally, exploiting the full potential of the interplay between the three topics should allow us to solve fundamental open problems in these areas.