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Random walks in finite Abelian groups with Birkhoff subpolytopes of doubly stochastic matrices and their physical implementation

A. Vourdas

Abstract

Random walks in a finite Abelian group $G$ are studied. They use Markov chains with doubly stochastic transition matrices, in a Birkhoff subpolytope ${\cal B}(G)$ associated with the group $G$. It is shown that all future probability vectors belong to a polytope which does not depend on the transition matrices, and which shrinks during time evolution. Various quantities are used to describe the probability vectors: the majorization preorder, Lorenz values and the Gini index, entropic quantities, and the total variation distance. The general results are applied to the additive group ${\mathbb Z}(d)$, and to the Heisenberg-Weyl group $HW(d)/{\mathbb Z}(d)$. A physical implementation of random walks in ${\mathbb Z}(d)$ that involves a sequence of non-selective projective measurements, is discussed. A physical implementation of random walks in the Heisenberg-Weyl group $HW(d)/{\mathbb Z}(d)$ using a sequence of non-selective POVM measurements with coherent states, is also presented.

Random walks in finite Abelian groups with Birkhoff subpolytopes of doubly stochastic matrices and their physical implementation

Abstract

Random walks in a finite Abelian group are studied. They use Markov chains with doubly stochastic transition matrices, in a Birkhoff subpolytope associated with the group . It is shown that all future probability vectors belong to a polytope which does not depend on the transition matrices, and which shrinks during time evolution. Various quantities are used to describe the probability vectors: the majorization preorder, Lorenz values and the Gini index, entropic quantities, and the total variation distance. The general results are applied to the additive group , and to the Heisenberg-Weyl group . A physical implementation of random walks in that involves a sequence of non-selective projective measurements, is discussed. A physical implementation of random walks in the Heisenberg-Weyl group using a sequence of non-selective POVM measurements with coherent states, is also presented.
Paper Structure (21 sections, 4 theorems, 123 equations, 1 table)

This paper contains 21 sections, 4 theorems, 123 equations, 1 table.

Table of Contents

  1. Introduction
  2. Non-selective measurements on a system with finite dimensional Hilbert space
  3. Non-selective measurements with orthogonal projectors
  4. Non-selective POVM measurements with coherent states
  5. A single index notation for $HW(d)/{\mathbb Z}(d)\simeq {\mathbb Z}(d)\times {\mathbb Z}(d)$
  6. Polytopes of doubly stochastic matrices and polytopes of probability vectors
  7. Permutations and doubly stochastic matrices
  8. Probability vectors
  9. Birkhoff subpolytopes
  10. Polytopes of probability vectors
  11. Ergodic doubly stochastic matrices
  12. Quantities that describe probability vectors
  13. Sparsity of probability vectors: Majorization, Lorenz values and the Gini index
  14. Total variation distance between two probability vectors
  15. Entropic quantities
  16. ...and 6 more sections

Key Result

Proposition 3.7

Let ${\cal P}$ be a doubly stochastic matrix such that all eigenvalues of ${\cal P}$ apart from the eigenvalue $e_0=1$, have absolute value $|e_i|<1$. Then In this case we say that the doubly stochastic matrix is ergodic.

Theorems & Definitions (18)

  • Definition 3.1
  • Example 3.2
  • Example 3.3
  • Definition 3.4
  • Example 3.5
  • Example 3.6
  • Proposition 3.7
  • proof
  • Proposition 5.1
  • proof
  • ...and 8 more