Bell tests have become a powerful tool for checking security, quantifying randomness, detecting entanglement, and many other applications, as well as for investigating fundamental physical limits. Most Bell experiments make the assumptions of fair sampling and equal detection efficiency, and some also the assumption of setting reproducibility. Here, we point out that there are typical experimental imperfections and practices that lead to the violation of these assumptions and can go unnoticed. This is a problem that can invalidate the conclusions of many past and future Bell experiments and allow for malicious attacks. Detecting, quantifying, and fixing this problem is therefore of fundamental importance, especially for modern applications where the experimental values are used to reach quantitative conclusions about security, randomness, or entanglement. To illustrate the issue and its causes, we present a set of Bell experiments using polarization-entangled photons, where we identify common imperfections and practices that cause the failure of the assumptions. These experiments tell us which methods we should avoid and to which aspects of the setup we should pay special attention. We show that the failure of these assumptions results in a violation of a fundamental requirement in any Bell experiment, namely, non-signalling. We present a test based on the deviation from the non-signalling conditions that quantify and help us to fix the problem. We emphasize that adopting the measures and conducting the tests suggested here is necessary in order to obtain reliable conclusions in modern quantum information applications based on Bell tests.

Minimal informationally complete positive operator-valued measures (MIC-POVMs) are special kinds of measurement in quantum theory in which the statistics of their d(2)-outcomes are enough to reconstruct any d-dimensional quantum state. For this reason, MIC-POVMs are referred to as standard measurements for quantum information. Here, we report an experiment with entangled photon pairs that certifies, for what we believe is the first time, a MIC-POVM for qubits following a device-independent protocol (i.e., modeling the state preparation and the measurement devices as black boxes, and using only the statistics of the inputs and outputs). Our certification is achieved under the assumption of freedom of choice, no communication, and fair sampling.

Self-testing represents the strongest form of certification of a quantum system. Here, we theoretically and experimentally investigate self-testing of nonprojective quantum measurements. That is, how can one certify, from observed data only, that an uncharacterized measurement device implements a desired nonprojective positive-operator valued measure (POVM).We consider a prepare-and-measure scenario with a bound on the Hilbert space dimension and develop methods for (i) robustly self-testing extremal qubit POVMs and (ii) certifying that an uncharacterized qubit measurement is nonprojective. Our methods are robust to noise and thus applicable in practice, as we demonstrate in a photonic experiment. Specifically, we show that our experimental data imply that the implemented measurements are very close to certain ideal three- and four-outcome qubit POVMs and hence non-projective. In the latter case, the data certify a genuine four-outcome qubit POVM. Our results open interesting perspective for semi-device-independent certification of quantum devices.