No epistemic model can explain anti-distinguishability of quantum mixed preparations

We address the fundamental question of whether epistemic models can reproduce the empirical predictions of general quantum preparations. This involves comparing the quantum overlap determined by the anti-distinguishability of a set of mixed preparations with the epistemic overlap of the probability distribution over the ontic states describing these preparations. A set of quantum mixed states is deemed to be 'non-epistemic' when the epistemic overlap must be zero while the corresponding quantum overlap remains non-zero. In its strongest manifestation, a set of mixed quantum states is 'fully non-epistemic' if the epistemic overlap vanishes while the quantum overlap reaches its maximum value of one. Remarkably, we show that there exist sets of non-epistemic mixed states even in dimension 2, when the overlap between three mixed preparations is concerned. Moreover, we present quantum mixed states in dimensions 3 and 4 that are fully non-epistemic concerning the overlap between four and three preparations, respectively. We also establish a generic upper bound on the average ratio between the epistemic and quantum overlap for two mixed states. The ratio is shown to be arbitrarily small for certain pairs of quantum mixed states, signifying they are non-epistemic. All our findings are robustly testable in the prepare-and-measure experiments. In addition, we identify the instances where the existence of non-epistemic mixed states leads to the refutation of ψ-epistemic models and further note that some of the examples obtained here indeed fall into this category. Interestingly, any proof of preparation contextuality implies that the respective mixed states are non-maximally epistemic, a weaker version of non-epistemic where the epistemic overlap is required to be less than the quantum overlap.