Quantum stochastic Lie-Trotter product formula II

TL;DR

This paper extends the quantum stochastic Lie-Trotter product formula to a broader class of cocycles, introduces a quantum Ito algebra, and explores generator decompositions and extensions to quadratic forms, advancing quantum stochastic analysis.

Contribution

It proves a quantum stochastic Lie-Trotter product formula without the strong independence assumption, develops a quantum Ito algebra, and extends the framework to quadratic form generators.

Findings

Established a quantum stochastic Lie-Trotter product formula for quasicontractive cocycles.

Developed a quantum Ito algebra facilitating analysis of quantum stochastic processes.

Proposed a canonical decomposition of quantum stochastic generators and extended the algebra to quadratic forms.

Abstract

A natural counterpart to the Lie-Trotter product formula for norm-continuous one-parameter semigroups is proved, for the class of quasicontractive quantum stochastic operator cocycles whose expectation semigroup is norm continuous. Compared to previous such results, the assumption of a strong form of independence of the constituent cocycles is overcome. The analysis is facilitated by the development of some quantum Ito algebra. It is also shown how the maximal Gaussian component of a quantum stochastic generator may be extracted - leading to a canonical decomposition of such genarators, and the connection to perturbation theory is described. Finally, the quantum Ito algebra is extended to quadratic form generators, and a conjecture is formulated for the extension of the product formula to holomorphic quantum stochastic cocycles.