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Large deviation principle for moment map estimation

Alonso Botero, Matthias Christandl, Péter Vrana

Abstract

We consider a family of positive operator valued measures associated with representations of compact connected Lie groups. For many independent copies of a single state and a tensor power representation we show that the observed probability distributions converge to the value of the moment map. For invertible states we prove that the measures satisfy the large deviation principle with an explicitly given rate function.

Large deviation principle for moment map estimation

Abstract

We consider a family of positive operator valued measures associated with representations of compact connected Lie groups. For many independent copies of a single state and a tensor power representation we show that the observed probability distributions converge to the value of the moment map. For invertible states we prove that the measures satisfy the large deviation principle with an explicitly given rate function.

Paper Structure

This paper contains 14 sections, 21 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Related work.
  3. Preliminaries
  4. Complexification
  5. Moment map
  6. Borel subgroups
  7. Positive operator valued measures
  8. Moment map estimation
  9. The measurements
  10. Extension of the coadjoint action
  11. Rate function and some properties
  12. Upper bound
  13. Law of large numbers
  14. Lower bound

Key Result

Theorem 1.1

Let $\mathcal{H}$ be a finite dimensional Hilbert space, $K$ a compact connected group, $\pi:K\to U(\mathcal{H})$ and $\mu_m$ as above.

Theorems & Definitions (61)

  • Theorem 1.1: Large deviation principle
  • Theorem 1.2: Law of large numbers
  • Example 2.1
  • Example 2.2
  • Remark 2.3
  • Remark 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • ...and 51 more