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Continuous operations on non-Markovian processes

Fabio Costa, Jing Yang

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 generalize 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 a natural definition of Markovianity in continuous time. We illustrate the formalism by analyzing continuous measurements in a generalized Caldeira-Leggett model, demonstrating its applicability to realistic non-Markovian scenarios.

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 generalize 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 a natural definition of Markovianity in continuous time. We illustrate the formalism by analyzing continuous measurements in a generalized Caldeira-Leggett model, demonstrating its applicability to realistic non-Markovian scenarios.

Paper Structure

This paper contains 19 sections, 181 equations, 1 figure.

Table of Contents

  1. Conventions
  2. Review of coherent-state and phase-space path integrals
  3. Coherent-state representation
  4. Connection with the phase-space representation
  5. Derivation of the coherent-state functional Born rule \ref{['eq:functionalBorn']} in the main text.
  6. Properties of the process and operation functionals
  7. Positivity
  8. Causality for the closed-future process functional $\mathcal{W}$ and the open-future process functional $\widetilde{\mathcal{W}}$
  9. Causality for discrete-time process
  10. Causality for the continuous-time processes
  11. Causality, trace-preserving property for the open-boundary process functionals $\underset{\widetilde{\space}}{\mathcal{W}}$ and the definition of Markovianity
  12. Recover the discrete-time formalism
  13. The Born rule
  14. Recovering discrete-time properties
  15. Position-space path integrals
  16. ...and 4 more sections

Figures (1)

  • Figure 1: (a) Schematic illustration of discrete processes and operations. Operations act from even to odd time while the process takes over from odd to even times. (b) Trotterization of the joint evolution of the meter, system and environment. Upon taking the continuous-time limit, we obtain the representation of the continuous-time quantum processes and testers respectively.