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arXiv:2603.08431

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

Apostol Vourdas

Published Mar 9, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2603.08431

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Random walks in a finite Abelian group GGG are studied. They use Markov chains with doubly stochastic transition matrices, in a Birkhoff subpolytope B(G){\cal B}(G)B(G) associated with the group GGG. 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 Z(d){\mathbb Z}(d)Z(d), and to the Heisenberg-Weyl group HW(d)/Z(d)HW(d)/{\mathbb Z}(d)HW(d)/Z(d). A physical implementation of random walks in Z(d){\mathbb Z}(d)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)/Z(d)HW(d)/{\mathbb Z}(d)HW(d)/Z(d) using a sequence of non-selective POVM measurements with coherent states, is also presented.
Abstract Random walks in a finite Abelian group G are studied. They use Markov chains with doubly stochastic transition matrices, in a Birkhoff subpolytope 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 Z ( d ) , and to the Heisenberg–Weyl group H W ( d ) / Z ( d ) . A physical implementation of random walks in Z ( d ) that involves a sequence of non-selective projective measurements, is discussed. A physical implementation of random walks in the Heisenberg–Weyl group H W ( d ) / Z ( d ) using a sequence of non-selective positive operator-valued measure measurements with coherent states, is also presented.
A time-dependent finite-state Markov chain that uses doubly stochastic transition matrices, is considered. Entropic quantities that describe the randomness of the probability vectors, and also the randomness of the discrete paths, are studied. Universal convex polytopes are introduced which contain all future probability vectors, and which are based on the Birkhoff-von Neumann expansion for doubly stochastic matrices. They are universal in the sense that they depend only on the present probability vector, and are independent of the doubly stochastic transition matrices that describe time evolution in the future. It is shown that as the discrete time increases these convex polytopes shrink, and the minimum entropy of the probability vectors in them increases. These ideas are applied to a sequence of non-selective measurements (with different projectors in each step) on a quantum system with ddd-dimensional Hilbert space. The unitary time evolution in the intervals between the measurements, is taken into account. The non-selective measurements destroy stroboscopically the non-diagonal elements in the density matrix. This `hermaphrodite' system is an interesting combination of a classical probabilistic system (immediately after the measurements) and a quantum system (in the intervals between the measurements). Various examples are discussed. In the ergodic example, the system follows asymptotically all discrete paths with the same probability. In the example of rapidly repeated non-selective measurements, we get the well known quantum Zeno effect with `frozen discrete paths' (presented here as a biproduct of our general methodology based on Markov chains with doubly stochastic transition matrices).
A time-dependent finite-state Markov chain that uses doubly stochastic transition matrices, is considered. Entropic quantities that describe the randomness of the probability vectors, and also the randomness of the discrete paths, are studied. Universal convex polytopes are introduced which contain all future probability vectors, and which are based on the Birkhoff–von Neumann expansion for doubly stochastic matrices. They are universal in the sense that they depend only on the present probability vector, and are independent of the doubly stochastic transition matrices that describe time evolution in the future. It is shown that as the discrete time increases these convex polytopes shrink, and the minimum entropy of the probability vectors in them increases. These ideas are applied to a sequence of non-selective measurements (with different projectors in each step) on a quantum system with d-dimensional Hilbert space. The unitary time evolution in the intervals between the measurements, is taken into account. The non-selective measurements destroy stroboscopically the non-diagonal elements in the density matrix. This ‘hermaphrodite’ system is an interesting combination of a classical probabilistic system (immediately after the measurements) and a quantum system (in the intervals between the measurements). Various examples are discussed. In the ergodic example, the system follows asymptotically all discrete paths with the same probability. In the example of rapidly repeated non-selective measurements, we get the well known quantum Zeno effect with ‘frozen discrete paths’ (presented here as a biproduct of our general methodology based on Markov chains with doubly stochastic transition matrices).
A time-dependent finite-state Markov chain that uses doubly stochastic transition matrices, is considered. Entropic quantities that describe the randomness of the probability vectors, and also the randomness of the discrete paths, are studied. Universal convex polytopes are introduced which contain all future probability vectors, and which are based on the Birkhoff–von Neumann expansion for doubly stochastic matrices. They are universal in the sense that they depend only on the present probability vector, and are independent of the doubly stochastic transition matrices that describe time evolution in the future. It is shown that as the discrete time increases these convex polytopes shrink, and the minimum entropy of the probability vectors in them increases. These ideas are applied to a sequence of non-selective measurements (with different projectors in each step) on a quantum system with d-dimensional Hilbert space. The unitary time evolution in the intervals between the measurements, is taken into account. The non-selective measurements destroy stroboscopically the non-diagonal elements in the density matrix. This ‘hermaphrodite’ system is an interesting combination of a classical probabilistic system (immediately after the measurements) and a quantum system (in the intervals between the measurements). Various examples are discussed. In the ergodic example, the system follows asymptotically all discrete paths with the same probability. In the example of rapidly repeated non-selective measurements, we get the well known quantum Zeno effect with ‘frozen discrete paths’ (presented here as a biproduct of our general methodology based on Markov chains with doubly stochastic transition matrices).
Algebraic random walks (ARW) and quantum mechanical random walks (QRW) are investigated and related. Based on minimal data provided by the underlying bialgebras of functions defined on e. g the real line R, the abelian finite group Z_N, and the canonical Heisenberg-Weyl algebra hw, and by introducing appropriate functionals on those algebras, examples of ARWs are constructed. These walks involve short and long range transition probabilities as in the case of R walk, bistochastic matrices as for the case of Z_N walk, or coherent state vectors as in the case of hw walk. The increase of classical entropy due to majorization order of those ARWs is shown, and further their corresponding evolution equations are obtained. Especially for the case of hw ARW, the diffusion limit of evolution equation leads to a quantum master equation for the density matrix of a boson system interacting with a bath of quantum oscillators prepared in squeezed vacuum state. A number of generalizations to other types of ARWs and some open problems are also stated. Next, QRWs are briefly presented together with some of their distinctive properties, such as their enhanced diffusion rates, and their behavior in respect to the relation of majorization to quantum entropy. Finally, the relation of ARWs to QRWs is investigated in terms of the theorem of unitary extension of completely positive trace preserving (CPTP) evolution maps by means of auxiliary vector spaces. It is applied to extend the CPTP step evolution map of a ARW for a quantum walker system into a unitary step evolution map for an associated QRW of a walker+quantum coin system. Examples and extensions are provided.
We investigate the use of discrete-time quantum walks to sample from an almost-uniform distribution, in the absence of any external source of randomness. Integers are encoded on the vertices of a cycle graph, and a quantum walker evolves for a fixed number of steps before its position is measured and recorded. The walker is then reset to the measured site, and the procedure is iterated to produce the sequence of random numbers. We show that when the quantum walk parameters, such as the coin operator and initial state, satisfy the conditions of the ergodic theorem for random walks on finite groups, the resulting sequence converges asymptotically to the uniform distribution. Although correlations between successive outcomes are unavoidable, they can be significantly reduced by a suitable choice of the evolution time. By analyzing the iterated convolution of the quantum walk transition probability and exploiting the ergodic theorem, we demonstrate convergence of the marginal distributions toward the uniform distribution in the asymptotic limit.