Geometric Probabilities and Fibonacci Numbers for Maximally Random n-Qubit Quantum Information States

Abstract

The problems of Hadamard quantum coin flipping in n-trials and related generalized Fibonacci sequences of numbers were introduced in [1]. It was shown that for an arbitrary number of repeated consecutive states, probabilities are determined by Fibonacci numbers for duplicated states, Tribonacci numbers for triplicated states and N-Bonacci numbers for arbitrary N-plicated states. In the present paper we generalize these results for direct product of multiple qubit states and arbitrary position of repeated states. The calculations are based on structure of Fibonacci trees in space of qubit states, growing in the left and in the right directions, and number of branches and allowed paths on the trees. By using $n$-qubit quantum coins as random n-qubit states with maximal Shannon entropy, we show that quantum probabilities can be calculated by means of geometric probabilities. It illustrates possible application of geometric probabilities in quantum information theory. The Golden ratio of probabilities and the limit of n going to infinity are discussed.