Random walks in finite Abelian groups with Birkhoff subpolytopes of doubly stochastic matrices and their physical implementation
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866908873732915200 |
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| author | Vourdas, A. |
| author_facet | Vourdas, A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08431 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Random walks in finite Abelian groups with Birkhoff subpolytopes of doubly stochastic matrices and their physical implementation Vourdas, A. Mathematical Physics 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. |
| title | Random walks in finite Abelian groups with Birkhoff subpolytopes of doubly stochastic matrices and their physical implementation |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2603.08431 |