Quantum Probability and the Born Ensemble

Abstract

We formulate a discrete two-state stochastic process with elementary rules that give rise to Born statistics and reproduce the probabilities from the Schr\"odinger equation under an associated Hamiltonian matrix, which we identify. We define the probability to observe a state, classical or quantum, in proportion to the number of \textit{events} at that state--number of ways a walker may materialize at a point of observation at time t through a sequence of transitions starting from known initial state at t=0. The quantum stochastic process differs from its classical counterpart in that the quantum walker is a pair of qubits, each transmitted independently through all possible paths to a point of observation, and whose recombination may produce a positive or negative event (classical events are never negative). We represent the state of the walker via a square matrix of recombination events, show that the indeterminacy of the qubit state amounts to rotations of this matrix, and show that the Born rule counts the number qubits on this matrix that remain invariant over a full rotation.