From the Heisenberg to the Schr\"{o}dinger Picture: Quantum Stochastic Processes and Process Tensors

TL;DR

This paper explores the relationship between quantum stochastic processes and process tensors, showing how the latter can be derived from the former using extended correlation kernels, bridging a gap in quantum probability theory.

Contribution

It provides a detailed exposition connecting quantum stochastic processes with the process tensor formalism, clarifying their relationship within quantum probability theory.

Findings

Process tensor formalism emerges from quantum stochastic processes.

Extended correlation kernels incorporate ancillas to relate the two frameworks.

The paper bridges a gap between quantum probability theory and process tensor approaches.

Abstract

A general theory of quantum stochastic processes was formulated by Accardi, Frigerio and Lewis in 1982 within the operator-algebraic framework of quantum probability theory, as a non-commutative extension of the Kolmogorovian classical stochastic processes. More recently, studies on non-Markovian quantum processes have led to the discrete-time process tensor formalism in the Schr\"{o}dinger picture to describe the outcomes of sequential interventions on open quantum systems. However, there has been no treatment of the relationship of the process tensor formalism to the quantum probabilistic theory of quantum stochastic processes. This paper gives an exposition of quantum stochastic processes and the process tensor and the relationship between them. In particular, it is shown how the latter emerges from the former via extended correlation kernels incorporating ancillas.