# A class of stochastic products on the convex set of quantum states

## TL;DR

This paper introduces stochastic products on quantum states, explores their properties, especially group-covariant and associative types, and demonstrates how they form Banach algebras, with implications for quantum symmetry and algebraic structures.

## Contribution

It develops a new class of group-covariant, associative stochastic products called twirled stochastic products, linking them to projective representations and Banach algebra structures.

## Key findings

- Twirled stochastic products form Banach algebras extending to trace class operators.
- These products are commutative when based on abelian groups.
- Explicit treatment of Weyl system-generated stochastic products.

## 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 group-covariant, associative stochastic products, the so-called 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 so-called 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.