Quantum and Classical Probability with Applications to Physics and Biology
Research focus
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1. Quantum stochastic calculus. This is an Ito type stochastic calculus with non-commuting noises, used in quantum probability to construct various types of quantum stochastic processes and in physics to build up models of open systems, mainly in quantum optics. a. The theory of quantum stochastic differential equations (quantum SDE’s) [J5, BC34, BC38] b. Applications to the theory of classical SDE’s [P33] c. Hamiltonian evolutions and quantum SDE’s [J19, J23] d. Connections with operator theory, C*-algebras, quantum stochastic processes [J7, J25, J26] 2. Quantum Markov semigroups, quantum flows and applications to open quantum systems. Quantum Markov semigroups are a non-commutative analogue of the transition semigroups in the theory of classical Markov processes and were introduced in the sixties/seventies to give the dynamics of open quantum systems without memory (quantum master equations). Quantum (Markov) flows are a non-commutative analogue of classical Markov processes. a. General theory and classification [J5, J6, J17, P27, P29, P32, BC37] b. Study of special models [J15, J20, P30] c. Applications [J20, BC36] 3. The theory of quantum continual measurements. This is the theory which describes the observation of a quantum system continually in time: the observed system is a quantum one, but the output of the observation is a classical stochastic process. The devepments of this theory need operator valued measures, classical and quantum SDE’s, filtering theory. [BC36] a. Quantum trajectories (classical SDE’s in quantum continual measurement theory) [J1, J11, BC35] b. Detection theory (quantum SDE’s in quantum continual measurement theory) [BC36] 4. Quantum information. This rapidly growing new field treats all aspects of information transmission and processing via quantum systems and puts together information theory, quantum measurement theory, statistical properties of quantum theories… a. Entropies and bounds on information transmission in quantum systems [J11, J12, J21, P31, BC35] b. Error correction in quantum information transmission and storage [J2, J9] c. Quantum algorithms [J20] 5. Interacting particle systems. Interacting particles systems [J10,J13,J14,J18] can be used to model many situation in which the object of study is a "thermodynamic-like" system like a real thermodynamic model (classical or quantum), but also like a biological population or a complex economic system. An interacting particle system is a stochastic process taking values in a product space. Every coordinate of the product space represents the state of a single molecule or individual. Every single molecule or individual interact with others in some way. The typical problem in this field is the description, with full mathematical rigour, of the global behaviour of the system starting from a given "microscopic" law (the interaction between molecules or individuals). 69 a. Relaxation time for interacting particle systems [J10,J13,J14,J18,J22]. b. Probability of survival for biological populations (work in progress – accepted: http://www.imstat.org/aap/future_papers.html).
Dipartimento di afferenza
Docenti afferenti
Full Professors
Barchielli Alberto
Fagnola Franco
Associate Professors
Posta Gustavo
Assistant Professors
Gregoratti Matteo
Zucca Fabio