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Martingale Projections and Quantum Decoherence

Lane P. Hughston, Levent A. Mengütürk

TL;DR

Open quantum systems and decoherence are studied through a new class of projections called supermartingale projections, defined as boundedness-preserving endomorphisms on path-valued random variables and linked to conditional expectations. The framework uses path transformations on a double-union of Polish path spaces and formalizes projection classes as negative, positive, and neutral projection families to generalize classical martingales. The main results show that a supermartingale projection on the diagonal magnitudes of the density matrix induces decoherence, a submartingale projection induces information gain (Shannon-Wiener information increases in expectation), and a martingale projection entails both, with these effects localized to time compartments if needed. Together, these findings connect classical martingale theory to quantum state evolution in open systems and offer a principled way to analyze energy-information trade-offs in quantum measurement and decoherence.

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.

Martingale Projections and Quantum Decoherence

TL;DR

Open quantum systems and decoherence are studied through a new class of projections called supermartingale projections, defined as boundedness-preserving endomorphisms on path-valued random variables and linked to conditional expectations. The framework uses path transformations on a double-union of Polish path spaces and formalizes projection classes as negative, positive, and neutral projection families to generalize classical martingales. The main results show that a supermartingale projection on the diagonal magnitudes of the density matrix induces decoherence, a submartingale projection induces information gain (Shannon-Wiener information increases in expectation), and a martingale projection entails both, with these effects localized to time compartments if needed. Together, these findings connect classical martingale theory to quantum state evolution in open systems and offer a principled way to analyze energy-information trade-offs in quantum measurement and decoherence.

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.

Paper Structure

This paper contains 6 sections, 5 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Path Transformations
  3. Martingale Projections
  4. Quantum Decoherence
  5. Conclusion
  6. Appendix

Key Result

Proposition 2.5

Let $\mathcal{I} = \{1,\ldots,K\}$ be an ordered set for an integer $K\geq 2$ and let $\boldsymbol{\Gamma}^{\left(\alpha^{(i)},\beta^{(i)}\right)\mapsto\left(\alpha^{(i+1)},\beta^{(i+1)}\right)}$ be a path transformation for every $i\in\mathcal{I}\setminus\{K\}$. Then, for any $\alpha^{(i)}\leq \bet is a path transformation on $\boldsymbol{\Lambda}(\mathbb M \,, \mathbb{T})$ from $\mathbb{T}_{[\a

Theorems & Definitions (26)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Example 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Remark 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 16 more