Non-abelian Weyl Commutation Relations and the Series Product of Quantum Stochastic Evolutions

TL;DR

This paper reveals that the series product in quantum stochastic models is a non-abelian extension of the Weyl commutation relations, connecting algebraic rules with the Heisenberg group.

Contribution

It establishes a novel link between the series product and non-abelian Weyl relations, providing a Lie-Trotter formula for combining quantum stochastic generators.

Findings

Series product relates to the Heisenberg group.

Generalizes Weyl commutation relations to non-abelian case.

Provides a Lie-Trotter product formula for quantum evolutions.

Abstract

We show that the series product, which serves as an algebraic rule for connecting state-based input/output systems, is intimately related to the Heisenberg group and the canonical commutation relations. The series product for quantum stochastic models then corresponds to a non-abelian generalization of the Weyl commutation relation. We show that the series product gives the general rule for combining the generators of quantum stochastic evolutions using a Lie-Trotter product formula.