Fig. 1

Perturbed coin. The process that we study here is a coin with two outcomes, 0 and 1. The transition probabilities, T_ij, between different outcomes are determined by l and m for i, j ∈ {0, 1}. The optimal classical model uses the causal states ({S_i}) depicted in the circles. There is a simple mapping from the past of the process to the relevant causal state: the last outcome from the coin determines the input causal state. Arrows, with the associated expressions j|T_ij, represent the transitions from causal states S_i to S_j with probability T_ij, emitting the classical outcome j. In the quantum model, the causal states become quantum states, {|S_i〉}