Brownian motion on Lie groups and open quantum systems

Abstract

We study the twirling semigroups of (super)operators, namely, certain quantum dynamical semigroups that are associated, in a natural way, with the pairs formed by a projective representation of a locally compact group and a convolution semigroup of probability measures on this group. The link connecting this class of semigroups of operators with (classical) Brownian motion is clarified. It turns out that every twirling semigroup associated with a finite-dimensional representation is a random unitary semigroup, and, conversely, every random unitary semigroup arises as a twirling semigroup. Using standard tools of the theory of convolution semigroups of measures and of convex analysis, we provide a complete characterization of the infinitesimal generator of a twirling semigroup associated with a finite-dimensional unitary representation of a Lie group.

32 pages

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Cited by in corpus (8)

• Dynamical Vector Fields on the Manifold of Quantum States
• Characterizing the dynamical semigroups that do not decrease a quantum entropy
• Symmetry witnesses
• Operators versus functions: from quantum dynamical semigroups to tomographic semigroups
• Dip-ramp-plateau for Dyson Brownian motion from the identity on $U(N)$
• A class of stochastic products on the convex set of quantum states
• Discovering the manifold facets of a square integrable representation: from coherent states to open systems
• On a certain class of semigroups of operators