Quantum mechanics is
deterministic and probabilistic by nature, but until
recently tools from stochastic processes are surprisingly
underused in quantum mechanics. The situation has however
rapidly changed in recent years. Experimental progresses
in realizing stable and controllable quantum systems gave
new impetus to study unexplored territory of quantum
dynamics and it provides a unique opportunity to develop
and test new mathematical ideas dealing with open quantum
systems. New approaches rooted in probability theory have
thus emerged. To quote but a few: quantum noise theory
applied to out-of-equilibrium quantum dynamics, random
matrix theory in quantum information theory, quantum
trajectories and quantum stochastic differential equations
applied to open and controllable quantum systems, quantum
random walks and their use for quantum algorithms, etc.
This remarkable blend of mathematical and physical ideas
is at the root of the extraordinary efficiency that
characterizes this scientific area. It is of growing
practical importance and at the same time provides a vital
source of fresh ideas and inspirations for those working
in more abstract directions.

Our project aims at
systematizing and developing the use of stochastic tools
in modern quantum physics. Our team gathers mathematicians
and theoretical physicists who played noticeable roles in
recent advances in stochastic methods applied to quantum
mechanics and to open quantum systems. We wish to develop
synergies to tackle challenging problems of modern quantum
physics using probabilistic approaches.

Our objectives are centered
on:

(a) Quantum noises and open
quantum systems;

(b) Quantum trajectories;

(c) Random states and random
channels in quantum information theory;

(d) Open quantum random
walks.

The following results are
among our aims:

-- We wish to obtain the
very first rigorous results about out-of-equilibrium open
quantum systems by a systematic use of quantum noises and
repeated quantum interactions. As done in the past twenty
years in classical statistical mechanics with classical
Langevin equations, modeling effects of quantum heat baths
by quantum Langevin equations should open the door for
tractable mathematical models encoding quantum
dissipation.

-- We want to develop what
certainly constitutes one of the most original and
powerful approach to recent conjectures in quantum
information theory: the use of random matrices, free
probability and operator algebraic tools. All the members
of this project who are involved in quantum information
theory are pioneers of this line of research and they did
obtain important and recognized results.

-- We want to develop the
mathematical foundations of quantum trajectories as well
as their domain of applications. Quantum trajectories are
instrumental in analyzing fundamental physical experiments
but, because of the difficult techniques from stochastic
calculus and stochastic control theory they involve, they
constitute important mathematical challenges.

-- We wish to adapt tools of
strongly interacting quantum systems to deal with
out-of-equilibrium mesoscopic systems. Cross-fertilization
between those tools and quantum noise theory leads us to
look for theoretical and mathematical formulations of
out-of-equilibrium low dimensional quantum systems and
their applications to control theory.

-- We want to develop the
applications of open quantum random walks. Some of us have
recently been involved in the emergence of a new promising
kind of quantum random walks, called open quantum random
walks, which take dissipation into account. We hope they
provide powerful tools to obtain tractable models of
out-of-equilibrium systems and we look forward to use them
to define quantum analogue of exclusion processes.
Deciphering their underlying (non-commutative) geometry is
one of our more speculative objectives.

Stéphane Attal : Head of the project and coordinator of Partner 1 (Lyon), Professeur at Institut Camille Jordan, University Lyon 1.

Denis Bernard : Coordinator of Partner 2 (Paris), Directeur de Recherche C.N.R.S. at Laboratoire de Physique Théorique, E.N.S Paris

Guillaume Aubrun : Maître de Conférences, Institut Camille Jordan, University Lyon 1

Michel Bauer : Directeur de Recherche, C.E.A. Saclay

Benoit Collins : Associate Professor, Department of Mathematics, Kyoto University (Japan)

Ion Nechita : Chargé de Recherches C.N.R.S., Laboratoire de Physique Théorique, Toulouse

Yan Pautrat : Maître de Conférences, Département de Mathématiques, Orsay University

Clément Pellegrini : Maître de Conférences, Laboratoire de Statistiques et Probabilités, Toulouse University

Ivan Bardet : PhD Student, Institut Camille Jordan, University Lyon 1

Tristan Benoist : Post-Doc

Julien Deschamps : Post-Doc, Department of Mathematics, University of Genova (Italy)

Antoine Tilloy : PhD Student, Laboratoire de Physique Théorique, E.N.S. Paris.