Not-Post-Peierls compatibility under noisy channels

The Pusey-Barrett-Rudolph (PBR) theorem deals with the realism of the quantum states. It establishes that every pure quantum state is real, in the context of quantum ontological models. Specifically, by guaranteeing the property of not-Post-Peierls ($\neg$PP) compatibility (or antidistinguishability) for a particular set of states $P$, together with the ad hoc postulate known as Preparation Independence Postulate (PIP), the theorem establishes that these two properties imply the $\psi$-onticity (realism) of the set of all pure states. This PBR result has triggered two particular lines of research: On the one hand, it has been possible to derive similar results without the use of the PIP, although at the expense of implying weaker properties than $\psi$-onticity. On the other hand, it has also been proven that the property of $\neg$PP compatibility alone is an explicit witness of usefulness for the task known as conclusive exclusion of states. In this work, we explore the $\neg$PP compatibility of the set of states $P$, when $P$ is under the interaction of some noisy channels, which would consequently let us identify some noisy scenarios where it is still possible to perform the task of conclusive exclusion of states. Specifically, we consider the set $P$ of $n$-qubit states in interaction with an environment by means of i) individual and ii) collective couplings. In both cases, we analytically show that the phenomenon of achieving $\neg$PP compatibility, although reduced, it is still present. Searching for an optimisation of this phenomenon, we report numerical experiments up to $n=4$ qubits.