Speaker: **Demosthenes Ellinas** (Technical Univ. Crete)

Title: Phase Plane Operator Valued Probability Measures: Constructions and Random Evolution

Date/Time: Thursday, August 8, 2013, 16:30-18:00

Room: 118 Math. Sci. Building

Abstract: Operator valued measures (OVM) are introduced on symplectic phase plane (PP), by means of quasi-probability Wigner function. Employing the metaplectic group MSp(2), constructions of such OVM for various PP regions (a.k.a region operators), are carried out. Stochastic increments, operating at the level of Wigner function/region operator, are then introduced algebraically via a commutative/co-commutative Hopf algebra of PP functions, together with a shift invariant functional. Dually we use completely positive trace preserving maps, operating at the level of PP state-density operator. In this way an algebraic random walk and its associated classical random walk is derived. Its asymptotic limit is shown to lead to a quantum master equation for the density operator or dually to a generalized diffusion equation for Wigner function. Analogous constructions with OVMs endowed with positivity (POVM), are worked out based on the representation theory of group ISO(2), which offers common ground for comparing the ensuing classical, algebraic and quantum random walks.