Quantum measurements are crucial for quantum technologies and give rise to some of the most classically counter-intuitive quantum phenomena. As such, the ability to certify the presence of genuinely non-classical joint measurements in a device- independent fashion is vital. However, previous work has either been non-device-independent or has relied on post-selection—the ability to discard all runs of an experiment in which a specific event did not occur. In the case of entanglement, the post-selection approach applies an entangled measurement to independent states and post-selects the outcome, inducing non-classical correlations between the states that can be device-independently certified using a Bell inequality. That is, it certifies measurement non-classicality not by what it is, but by what it does. This paper remedies this discrepancy by providing a novel notion of what measurement non-classicality is, which, in analogy with Bell’s theorem, corresponds to measurement statistics being incompatible with an underlying classical causal model. It is shown that this provides a more fine-grained notion of non-classicality than post- selection, as it certifies the presence of non-classicality that cannot be revealed by examining post-selected outcomes alone.
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