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State transfer on paths with weighted loops

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Abstract

We consider the fidelity of state transfer on an unweighted path on \(n \ge 3\) vertices, where a loop of weight w has been appended at each of the end vertices. It is known that if w is transcendental, then there is pretty good state transfer from one end vertex to the other; we prove a companion result to that fact, namely that there is a dense subset of \([1,\infty )\) such that if w is in that subset, pretty good state transfer between end vertices is impossible. Under mild hypotheses on w and t, we derive upper and lower bounds on the fidelity of state transfer between end vertices at readout time t. Using those bounds, we localise the readout times for which that fidelity is close to 1. We also provide expressions for, and bounds on, the sensitivity of the fidelity of state transfer between end vertices, where the sensitivity is with respect to either the readout time or the weight w. Throughout, the results rely on detailed knowledge of the eigenvalues and eigenvectors of the associated adjacency matrix.

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Acknowledgements

The authors are grateful to the anonymous referees, whose comments improved this article.

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  1. Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada

    Stephen Kirkland & Christopher M. van Bommel

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  1. Stephen Kirkland
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  2. Christopher M. van Bommel
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Correspondence to Christopher M. van Bommel.

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Stephen Kirkland: Research supported in part by The Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant RGPIN–2019–05408. Christopher M. van Bommel: Research supported in part by the Pacific Institute for the Mathematical Sciences (PIMS).

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Kirkland, S., van Bommel, C.M. State transfer on paths with weighted loops. Quantum Inf Process 21, 209 (2022). https://doi.org/10.1007/s11128-022-03558-x

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