Fibonacci–Lucas Recurrence-Indexed Quaternionic Unitaries: Channel Convergence, Clifford Proximity, and Controlled Entanglement
- 1Department of Mathematics, Faculty of Science, Kastamonu University, 37100 Kastamonu, Türkiye
- 2Department of Biomedical Engineering, Faculty of Engineering and Architecture, Kastamonu University, 37100 Kastamonu, Türkiye
Abstract
We investigate the normalized operator geometry of the mixed Fibonacci–Lucas quaternion Qn = Fn + Lni + Fn+1j + Ln+1k. Standard quaternion representations associate this sequence with single-qubit unitary channels. We first establish a convergence theorem for quaternion sequences with two distinct real characteristic roots and a strictly dominant positive root. It gives an exact average gate infidelity to the limit and sharp projective-step asymptotics, including the degenerate constant-channel case. For the Fibonacci–Lucas assignment, orthogonal Binet coefficient vectors yield exact finite-index formulas with projective steps of order φ−2n and average gate infidelities of order φ−4n. Strict comparison of the five projective Clifford rotation classes proves that the limiting unitary has a unique nearest projective Clifford, the cyclic Pauli-axis permutation, with average gate fidelity 115+5/90≈0.884408. For the controlled family, product-input maximization gives concurrence 1−an2, while independent Haar averaging gives linear-entropy entangling power 21−an2/9. Both admit explicit algebraic limits. Reproducible matrix and high-precision scalar calculations verify these results. The analysis supplies exact convergence, finite-group, and entanglement benchmarks for a recurrence-indexed family without assuming Hamiltonian dynamics.
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BibTeX
@article{unal2026fibonaccilucas,
title = {{Fibonacci–Lucas Recurrence-Indexed Quaternionic Unitaries: Channel Convergence, Clifford Proximity, and Controlled Entanglement}},
author = {Zafer Ünal and Can Doğan Vurdu},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203693},
url = {https://www.mdpi.com/2227-7390/14/20/3693},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Fibonacci–Lucas Recurrence-Indexed Quaternionic Unitaries: Channel Convergence, Clifford Proximity, and Controlled Entanglement
AU - Zafer Ünal
AU - Can Doğan Vurdu
PY - 2026
DA - 2026/10/12
JO - Mathematics
DO - 10.3390/math14203693
UR - https://www.mdpi.com/2227-7390/14/20/3693
AB - We investigate the normalized operator geometry of the mixed Fibonacci–Lucas quaternion Qn = Fn + Lni + Fn+1j + Ln+1k. Standard quaternion representations associate this sequence with single-qubit unitary channels. We first establish a convergence theorem for quaternion sequences with two distinct real characteristic roots and a strictly dominant positive root. It gives an exact average gate infidelity to the limit and sharp projective-step asymptotics, including the degenerate constant-channel case. For the Fibonacci–Lucas assignment, orthogonal Binet coefficient vectors yield exact finite-index formulas with projective steps of order φ−2n and average gate infidelities of order φ−4n. Strict comparison of the five projective Clifford rotation classes proves that the limiting unitary has a unique nearest projective Clifford, the cyclic Pauli-axis permutation, with average gate fidelity 115+5/90≈0.884408. For the controlled family, product-input maximization gives concurrence 1−an2, while independent Haar averaging gives linear-entropy entangling power 21−an2/9. Both admit explicit algebraic limits. Reproducible matrix and high-precision scalar calculations verify these results. The analysis supplies exact convergence, finite-group, and entanglement benchmarks for a recurrence-indexed family without assuming Hamiltonian dynamics.
SN - 2227-7390
ER -