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We provide an interesting two-party parity oblivious communication game whosesuccess probability is solely determined by the Bell expression. Theparity-oblivious condition in an operational quantum theory implies thepreparation non-contextuality in an ontological model of it. We find that theaforementioned Bell expression has two upper bounds in an ontological model;the usual local bound and a non-trivial preparation non-contextual boundarising from the non-trivial parity-oblivious condition, which is smaller thatthe local bound. We first demonstrate the communication game when both Aliceand Bob perform three measurements of dichotomic observables in theirrespective sites. The optimal quantum value of the Bell expression in thisscenario enables us to device-independently self-test the maximally entangledstate and trine-set of observables, three-outcome qubitpositive-operator-valued-measures (POVMs) and 1.58 bit of local randomness.Further, we generalize the above communication game in that both Alice and Bobperform the same but arbitrary (odd) number ($n> 3$) of measurements. Based onthe optimal quantum value of the relevant Bell expression for any arbitrary$n$, we have also demonstrated device-independent self-testing of state andmeasurements.

Oblivious communication game, self-testing of projective and non-projective measurements and certification of randomness