A Resolution of the Ito-Stratonovich Debate in Quantum Stochastic Processes

TL;DR

This paper introduces a novel quantum noise homogenization scheme that resolves the longstanding Ito--Stratonovich ambiguity in non-Markovian quantum stochastic processes driven by colored noise, establishing a consistent Markovian limit.

Contribution

The authors develop a phase-space augmentation and perturbative coarse-graining method to connect non-Markovian colored-noise quantum processes to their Markovian limits, clarifying the Ito--Stratonovich debate.

Findings

The Markovian limit corresponds to the Stratonovich convention with renormalized coefficients.

The approach explicitly derives effective Markovian generators for complex quantum noise.

It characterizes a family of physically relevant non-Markovian quantum processes.

Abstract

Quantum stochastic processes are widely used in describing open quantum systems and in the context of quantum foundations. Physically relevant quantum stochastic processes driven by multiplicative colored noise are generically non-Markovian and analytically intractable. Further, their Markovian limits are generically inequivalent when using either the Ito or Stratonovich conventions for the same quantum stochastic processes. We introduce a quantum noise homogenization scheme that temporally coarse-grains non-Markovian, colored-noise-driven quantum stochastic processes and connects them to their effective white-noise (Markovian) limits. Our approach uses a novel phase-space augmentation that maps the non-Markovian dynamics into a higher-dimensional Markovian system and then applies a controlled perturbative coarse-graining scheme in the characteristic time scales of the noise. This allows an explicit analytical algorithm to derive effective Markovian generators with renormalized coefficients and enables various physical constraints, such as causality, to be imposed on them. We thus resolve the Ito--Stratonovich ambiguity for multiplicative colored-noise-driven quantum stochastic processes, wherein we show that their consistent Markovian limit corresponds to the Stratonovich convention with renormalized coefficients as well as correction terms in Ito's convention. By assuming their Markovian limit unravels causal, completely positive and trace-preserving dynamics, we further characterize a physically relevant family of non-Markovian quantum stochastic processes driven by multiplicative colored~noise.