Quantum Coin Flipping, Qubit Measurement and Generalized Fibonacci Numbers

TL;DR

This paper explores quantum coin measurement processes using Fibonacci and generalized sequences, deriving probability formulas, generating functions, and entropies for various quantum systems including qubits, qutrits, and qudits.

Contribution

It introduces a novel framework linking quantum coin measurement outcomes to Fibonacci and N-Bonacci sequences, extending to higher-dimensional quantum systems.

Findings

Derived probability formulas in terms of Fibonacci and Lucas numbers.

Calculated the generating functions and Shannon entropy for quantum states.

Generalized results to qutrit and qudit systems using N-Bonacci sequences.

Abstract

The problem of Hadamard quantum coin measurement in $n$ trials, with 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 position of repeated states are derived in terms of Lucas and Fibonacci numbers. For generic qubit coin, the formulas are expressed by Fibonacci and more general, $N$-Bonacci polynomials in qubit probabilities. The generating function for probabilities, the Golden Ratio limit of these probabilities and Shannon entropy for corresponding states are determined. By generalized Born rule and universality of $n$-qubit measurement gate, we formulate problem in terms of generic $n$-qubit states and construct projection operators in 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.