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3 papers
Quantum coin flipping, qubit measurement, and generalized
Fibonacci numbers
O. K. Pashaev Department of Mathematics, Izmir Institute of Technology, Urla, Izmir, Turkey
Abstract:
The problem of Hadamard quantum coin measurement in
$n$ trials, with an arbitrary number of repeated consecutive last states, is formulated in terms of Fibonacci sequences for duplicated states, Tribonacci numbers for triplicated states, and
$N$-Bonacci numbers for arbitrary
$N$-plicated states. The probability formulas for arbitrary positions of repeated states are derived in terms of the Lucas and Fibonacci numbers. For a generic qubit coin, the formulas are expressed by the Fibonacci and more general,
$N$-Bonacci polynomials in qubit probabilities. The generating function for probabilities, the Golden Ratio limit of these probabilities, and the Shannon entropy for corresponding states are determined. Using a generalized Born rule and the universality of the
$n$-qubit measurement gate, we formulate the problem in terms of generic
$n$-qubit states and construct projection operators in a Hilbert space, constrained on the Fibonacci tree of the states. The results are generalized to qutrit and qudit coins described by generalized Fibonacci-
$N$-Bonacci sequences.
Keywords:
Fibonacci numbers, quantum coin, qubit, qutrit, qudit, quantum measurement, Tribonacci numbers, $N$-Bonacci numbers.
PACS:
03.67.-a
MSC: 81P45,
11B39 Received: 20.02.2021
Revised: 20.02.2021
DOI:
10.4213/tmf10078
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