Continuous operations on non-Markovian processes

Abstract

Continuous measurements are central to quantum control and sensing, yet lack a model-independent operational description that can be applied to arbitrary non-Markovian processes without specifying a microscopic measurement model. Existing multi-time frameworks, such as process matrices, allow for an arbitrary sequence of operations to be applied on a general process, but are restricted to interventions at discrete times and cannot represent measurements of finite duration. We introduce a continuous-time extension of multi-time quantum processes based on process and operation functionals, which generalise the Feynman-Vernon influence functional and yield a continuous Born rule that cleanly separates processes from operations. This framework provides a consistent representation of non-Markovian dynamics under continuous monitoring and leads to natural definitions of causality and Markovianity in continuous time, opening the way to formalising quantum causal structures in the continuum. We illustrate the formalism by analysing continuous measurements in a generalised Caldeira-Leggett model, demonstrating its applicability to realistic non-Markovian scenarios.

Expanded derivations and discussion. 9+25 pages, 2 figures, 225 equations