Quantum Probability and the Born Ensemble

Abstract

We formulate a discrete two-state stochastic model with elementary rules that give rise to Born statistics and reproduce the probabilities of the Schrodinger 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 events at that state--number of ways the walker may materialize at a point of observation at time t, 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. We represent the state of the walker via a square matrix of recombination events, interpret the indeterminacy of the qubit state as rotations of this matrix, and show that the Born rule counts the number elements on this matrix that remain invariant over a full rotation.Comment: 13 pages including appendi