A class of stochastic products on the convex set of quantum states
Abstract
We introduce the notion of stochastic product as a binary operation on the convex set of quantum states (the density operators) that preserves the convex structure, and we investigate its main consequences. We consider, in particular, stochastic products that are covariant wrt a symmetry action of a locally compact group. We then construct an interesting class of groupcovariant, associative stochastic products, the socalled twirled stochastic products. Every binary operation in this class is generated by a triple formed by a square integrable projective representation of a locally compact group, by a probability measure on that group and by a fiducial density operator acting in the carrier Hilbert space of the representation. The salient properties of such a product are studied. It is argued, in particular, that, extending this binary operation from the density operators to the whole Banach space of trace class operators, this space becomes a Banach algebra, a socalled twirled stochastic algebra. This algebra is shown to be commutative in the case where the relevant group is abelian. In particular, the commutative stochastic products generated by the Weyl system are treated in detail. Finally, the physical interpretation of twirled stochastic products and various interesting connections with the literature are discussed.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 July 2019
 DOI:
 10.1088/17518121/ab29be
 arXiv:
 arXiv:1903.11369
 Bibcode:
 2019JPhA...52D5302A
 Keywords:

 quantum state;
 group representation;
 operator algebra;
 quantum measurement;
 Mathematical Physics
 EPrint:
 42 pages