Fig. 1. Scenario with two participants. x: input message of Alice; y, b: input and output of Bob, respectively; ρ_x: state that Alice sends to Bob.

Fig. 2. Scenario with three participants. x: input message of Alice; y, b: input and output of Bob, respectively; ρ_x: state that Alice sends to Bob; ρ^y_x,_b: post-processing state that Bob sends to Charlie.

Fig. 3. Bounds on average success probability under 2 → 1 task. Short dashed line corresponds to the classical strategy. Dotted line, dotted and dashed line, and dotted and solid line correspond to the “unitary,” “mixed,” and “measure and prepare” projective strategies, respectively. Solid line corresponds to the general strategy with unsharp measurements. All of the bounds are tight.

Fig. 4. Correlations between the two success probabilities (P^B_succ, P^C_succ). The curve represents the boundary of the quantum set under 3 → 1; shadow rectangle represents classical boundary under 3 → 1.

Fig. 5. Bounds on average success probability under 3 → 1 task. Short dashed line corresponds to the classical strategy. Dotted line and dotted and dashed line correspond to the “unitary” and “measure and prepare” projective strategies, respectively. Solid line corresponds to the general strategy with unsharp measurements.