Robust certification of unsharp instruments through sequential quantum advantages in a prepare-measure communication game

Communication games are one of the widely used tools that are designed to demonstrate quantum supremacy over classical resources in which two or more parties collaborate to perform an information processing task to achieve the highest success probability of winning the game. We propose a specific two-party communication game in the prepare-measure scenario that relies on an encoding-decoding task of specific information. We first demonstrate that quantum theory outperforms the classical preparation noncontextual theory and the optimal quantum success probability of such a communication game enables the semi-device-independent certification of qubit states and measurements. Further, we consider the sequential sharing of quantum preparation contextuality and show that, at most, two sequential observers can share the quantum advantage. The suboptimal quantum advantages for two sequential observers form an optimal pair which certifies a unique value of the unsharpness parameter of the first observer. Since the practical implementation inevitably introduces noise, we devised a scheme to demonstrate the robust certification of the states and unsharp measurement instruments of both the sequential observers.

Figure 1

Sequential setting of the game.

Figure 2

Trade-off relation between the success probabilities achieved by Bob ${\mathrm{\Omega }}_B$ and Charlie ${\mathrm{\Omega }}_C$. The solid blue (dotted orange) line indicates the variation in success probability of Charlie with respect to Bob in classical (quantum) regimes. It is clear that by classical means no trade-off exists as there is no variation in Charlie's success probability with a change in Bob's success probability, whereas in quantum-mechanical prescriptions there is a solid trade-off in success probabilities between Charlie and Bob.

Figure 3

Variation between the average fidelity $\mathcal{F}$ for prepared states of Alice and the success probability ${\mathcal{A}}_B$ for a fixed value of the unsharpness parameter ${\eta }_B$.

Figure 4

Variation in the average fidelity of measurements of Bob with respect to success probability ${\mathcal{A}}_B$ for a fixed value of ${\eta }_B$.

Figure 5

Variation in the average fidelity of measurement with the variation of the success probability of Charlie.