Quantum coin flipping, qubit measurement, and generalized Fibonacci numbers

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.