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Decoherence as a high-dimensional geometrical phenomenon

Antoine Soulas

Abstract

We develop a mathematical formalism that allows to study decoherence with a great level generality, so as to make it appear as a geometrical phenomenon between reservoirs of dimensions. It enables us to give quantitative estimates of the level of decoherence induced by a purely random environment on a system according to their respectives sizes, and to exhibit some links with entanglement entropy.

Decoherence as a high-dimensional geometrical phenomenon

Abstract

We develop a mathematical formalism that allows to study decoherence with a great level generality, so as to make it appear as a geometrical phenomenon between reservoirs of dimensions. It enables us to give quantitative estimates of the level of decoherence induced by a purely random environment on a system according to their respectives sizes, and to exhibit some links with entanglement entropy.
Paper Structure (16 sections, 4 theorems, 37 equations, 2 figures)

This paper contains 16 sections, 4 theorems, 37 equations, 2 figures.

Table of Contents

  1. Keywords:
  2. The basics of decoherence
  3. First model: purely random environment
  4. Convergence to the uniform measure
  5. Most vectors are almost orthogonal
  6. Direct study of $\eta$
  7. Comparison with simulation and consequences
  8. Discussing the hypotheses
  9. Second model: interacting particles
  10. The environment feels the system
  11. Entanglement entropy as a measure of decoherence
  12. Alternative definitions for $\eta$
  13. Annex: decoherence estimated by the entanglement entropy with the environment
  14. Relation between $\eta$ and the level of classicality
  15. Relation between $\eta$ and the linear entropy
  16. ...and 1 more sections

Key Result

Proposition 2.1

We have $\lVert \nu_t - \mu \rVert_{TV} \underset{t \rightarrow +\infty} \longrightarrow 0$ exponentially fast. Moreover, if $T(\mathbb{S}^{n}) = \inf \{ t>0 \mid \lVert \nu_t - \mu \rVert_{TV} \leqslant \frac{1}{e} \}$ denotes the characteristic time to equilibrium for the brownian diffusion on $\m

Figures (2)

  • Figure 1: Simulation vs prediction for $d_{max}^{\varepsilon, s}(n)$
  • Figure 2: Simulation vs prediction for $d \mapsto \mathbb{E}(\eta_{n,d})$ at fixed $n$

Theorems & Definitions (10)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Theorem 2.5
  • Lemma 2.6
  • proof : Proof of Lemma
  • proof : Proof of Theorem \ref{['estimate eta']}