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In this work we relate notions of nonclassicality in the simplest nontrivialscenario (a prepare and measure scenario composed of four preparations and twobinary-outcome tomographically complete measurements). Specifically, we relatethe established method developed in [Pusey, PRA 98,022112(2018)] to witness aviolation of preparation noncontextuality, that is not suitable in experimentswhere the operational equivalences to be tested are specified in advance, witha novel approach based on the notion of bounded ontological distinctness forpreparations, defined in [Chaturvedi and Saha, Quantum 4, 345 (2020)]. In ourapproach, we test bounded ontological distinctness for two particularpreparations that are relevant in certain information processing tasks in thatthey are associated with the even and odd parity of the bits to communicate.When there exists an ontological model where this distance is preserved we talkof parity preservation. Our main result provides a noise threshold under whichviolating parity preservation (and so bounded ontological distinctness) agreeswith the established method for witnessing preparation contextuality in thesimplest nontrivial scenario. This is achieved by first relating the violationof parity preservation to the quantification of contextuality in terms ofinaccessible information as developed in [Marvian, arXiv:2003.05984(2020)],that we also show, given the way we quantify noise, to be more robust inwitnessing contextuality than Pusey's noncontextuality inequality. As anapplication of our findings, we treat the case of 2 bit parity-obliviousmultiplexing in the presence of noise. In particular, we provide a conditionfor which the result establishing preparation contextuality as a resource forthe quantum advantage of the protocol in the noiseless case still holds in thenoisy case.

Alternative robust ways of witnessing nonclassicality in the simplest scenario