A resolution of the Ito-Stratonovich debate in quantum stochastic processes

TL;DR

The paper tackles the ambiguity between Ito and Stratonovich calculi in quantum stochastic dynamics driven by colored noise. It develops a quantum noise homogenization method that augments the quantum state with the noise and uses a multi-scale coarse-graining to derive an analytically tractable Markovian limit. It shows that the Markovian limit consistent with CPTP maps is described by a Stratonovich-type stochastic Schrödinger equation with renormalized diffusion coefficients, accompanied by Ito corrections when reformulated in Ito form. This framework provides an explicit algorithm to compute the renormalized coefficients for a broad class of colored-noise driven quantum processes and clarifies when Stratonovich vs Ito is appropriate, with potential applications to open quantum systems, collapse models, and noise-driven many-body dynamics.

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 imposing various physical constraints 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 Ito correction terms. By assuming their Markovian limit unravels completely positive, trace-preserving maps, we further characterize a physically relevant family of non-Markovian quantum stochastic processes driven by multiplicative colored noise.