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Stochastic Methods in Quantum Mechanics – StoQ

Submission summary

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.

Project coordination

Stéphane Attal (Institut Camille Jordan, Université Lyon 1)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.


I.C.J. Institut Camille Jordan, Université Lyon 1
CNRS/ENS-Paris UMR 8549 Laboratoire de Physique Théorique

Help of the ANR 386,022 euros
Beginning and duration of the scientific project: September 2014 - 48 Months

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