Figure 4
Success probability \(P_s\) as a function of \(\theta\) for each pair of numbers \(\{a, b\}\) referee can give Alice and Bob for filling the row and column, respectively, with binary entries. Here, \(\theta\) represents the imperfection of the interaction between the logical qubit and the ancillary qubit, i.e. realizing not the desired \(CZ \equiv CP(\pi )\) but \(CP(\pi -\theta )\) operation. Operations \(A_1\) and \(B_1\) (corresponding to \(a=1\) and \(b=1\), respectively) are more complex than others that they contain more controlled operations, i.e. \(CP(\pi -\theta )\). Hence, success probability of players decreases faster for \(a=1\) or \(b=1\), and the fastest for \(a=b=1\).
