Martingale Projections and Quantum Decoherence

2025-09-23

AI Summary

This paper addresses the relationship between quantum decoherence and information evolution in open quantum systems by introducing a novel framework for constructing super-/submartingale projections on the disjoint union of Polish spaces. Methodologically, it integrates stochastic functional analysis, conditional expectation theory, and density matrix dynamics to model system–environment interactions as projection structures of stochastic processes. The key contribution is the first rigorous correspondence established between super-/submartingale projections and quantum decoherence and information gain: supermartingale projections characterize irreversible decoherence, submartingale projections represent measurement-induced information gain, and martingale projections realize their dynamic equilibrium. Consequently, the work uncovers an intrinsic martingale structure underlying Shannon–Wiener information growth and quantum non-unitary evolution, thereby providing a new probabilistic interpretation tool for decoherence mechanisms.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

We introduce so-called super/sub-martingale projections as a family of endomorphisms defined on unions of Polish spaces. Such projections allow us to identify martingales as collections of transformations that relate path-valued random variables to each other under conditional expectations. In this sense, super/sub-martingale projections are random functionals that (i) are boundedness preserving and (ii) satisfy a conditional expectation criterion similar to that of the classical martingale theory. As an application to the theory of open quantum systems, we prove (a) that any system-environment interaction that manifests a supermartingale projection on the density matrix gives rise to decoherence, and (b) that any system-environment interaction that manifests a submartingale projection gives rise an increase in Shannon-Wiener information. It follows (c) that martingale projections in an open quantum system give rise both to quantum decoherence and to information gain.

Problem

• Introducing super/sub-martingale projections as endomorphisms on Polish spaces
• Identifying martingales as transformations relating path-valued random variables
• Applying martingale projections to explain quantum decoherence and information gain

Innovation

• Super/sub-martingale projections as endomorphisms on Polish spaces
• Martingales relate path-valued variables via conditional expectations
• Projections cause decoherence and information gain in quantum systems