The Geometry of Transport in Quantum Walks and Parrondo's Paradox
Abstract
We study Parrondo's paradox -- the phenomenon where combining losing strategies yields a winning one -- in a minimal discrete-time quantum walk. We introduce the transport vector, encoded in the coin's steady state, whose inner product with the initial coin state gives the walker's asymptotic velocity. More generally, the paradox emerges exactly when the transport vector of the combined strategy falls outside the cone spanned by the individual ones -- a geometric criterion valid for any combination of strategies. It explains, for instance, why composing two coin operators within a single step can produce the paradox while simple alternation between them cannot, since alternation keeps the combined vector confined to the cone. The paradoxical set has nonzero measure, and we compute its probability explicitly in representative cases. This casts the paradox as one instance of designing reachable transport in quantum walks.
Related
- Parrondo's paradox for three discrete-time quantum walks with distinct shift operators
- Engineering entanglement and transport in interacting quantum walks with tailored potentials
- Scaling analysis of quantum geometry in second-order nonlinear transport
- Berry-Landau Fermi-liquid theory: transport in presence of quantum geometry
- Spectral Criterion for Disorder-Free Localization of Quantum Walks on Hypercube: QBN Approach
BibTeX
@misc{leon2026geometry,
title = {{The Geometry of Transport in Quantum Walks and Parrondo's Paradox}},
author = {Jose Alfredo de Leon and Mariana Pérez-Muralles and Jan Neuser and Carlos Pineda},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09269},
eprint = {2609.09269},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09269},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - The Geometry of Transport in Quantum Walks and Parrondo's Paradox
AU - Jose Alfredo de Leon
AU - Mariana Pérez-Muralles
AU - Jan Neuser
AU - Carlos Pineda
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09269
UR - https://arxiv.org/abs/2609.09269
AB - We study Parrondo's paradox -- the phenomenon where combining losing strategies yields a winning one -- in a minimal discrete-time quantum walk. We introduce the transport vector, encoded in the coin's steady state, whose inner product with the initial coin state gives the walker's asymptotic velocity. More generally, the paradox emerges exactly when the transport vector of the combined strategy falls outside the cone spanned by the individual ones -- a geometric criterion valid for any combination of strategies. It explains, for instance, why composing two coin operators within a single step can produce the paradox while simple alternation between them cannot, since alternation keeps the combined vector confined to the cone. The paradoxical set has nonzero measure, and we compute its probability explicitly in representative cases. This casts the paradox as one instance of designing reachable transport in quantum walks.
ER -