Construction and Rigorous Analysis of Quantum-Like States

Abstract: Extending upon the observations of the emergence of quantum-like states from classical complex synchronized networks, this work adds mathematical rigor to the analysis of single Quantum-Like (QL) bits constructed by eigenvectors of the adjacency matrices of such networks. First, we rigorously show that symmetric construction of such networks (regular undirected/symmetric bipartite graph $G_C$ connecting two regular undirected subgraphs $G_A,\,G_B$) leads to an equal superposition of the $|+\rangle, |-\rangle$ Hadamard states (with basis $|0\rangle,\,|1\rangle$ set from eigenvectors of the subgraphs), and provide an analysis of sufficient conditions on the network for construction of such states. Second, we prove two methods to construct any arbitrary single qubit state $|\psi\rangle = a|0\rangle + b|1\rangle,\, |a|^2+|b|^2=1$ and provide a switching lemma for the boundaries of both methods. The first method of construction is by detuning the regularities of the two subgraphs and the second is by asymmetrically constructing the bipartite connection matrix $C$ by allowing it to be directed, and then detuning those regularities. While the intuition is derived from the motivation of using complex synchronized networks for quantum information storage and computations, the proofs for constructing eigenvectors that interact in a quantum-like fashion only require the structure of the graph embedded in the adjacency matrix. Practically, this means that synchronization is not important to creating quantum-like bits, only that the edge weights are generally unit or close to unit and that the subgraphs are regular. As such, the results on combinations of random k-regular graphs (precisely Erd\H{o}s-R\'enyi graphs) may be independently interesting.