The Geometry of Transport in Quantum Walks and Parrondo's Paradox

• URL: http://arxiv.org/abs/2609.09269v1
• Date: Tue, 08 Sep 2026 18:00:00 GMT
• Title: The Geometry of Transport in Quantum Walks and Parrondo's Paradox
• Abstract summary: We study Parrondo's paradox in a minimal discrete-time quantum walk.<n>The paradox emerges exactly when the transport vector of the combined strategy falls outside the cone spanned by the individual ones.<n>This casts the paradox as one instance of designing reachable transport in quantum walks.
• Score: 0.034998703934432676
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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.

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