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Quantum measurements on a two-level system can have more than two independent outcomes, and in this case, the measurement cannot be projective. Measurements of this general type are essential to an operational approach to quantum theory, but so far, the nonprojective character of a measurement can only be verified experimentally by already assuming a specific quantum model of parts of the experimental setup. Here, we overcome this restriction by using a device-independent approach. In an experiment on pairs of polarization-entangled photonic qubits we violate by more than 8 standard deviations a Bell-like correlation inequality that is valid for all sets of two-outcome measurements in any dimension. We combine this with a device-independent verification that the system is best described by two qubits, which therefore constitutes the first device-independent certification of a nonprojective quantum measurement.
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 re
When transforming pairs of independent quantum operations according to the fundamental rules of quantum theory, an intriguing phenomenon emerges: some such higher-order operations may act on the input operations in an indefinite causal order. Recentl
Certifying the entanglement of quantum states with Bell inequalities allows one to guarantee the security of quantum information protocols independently of imperfections in the measuring devices. Here we present a similar procedure for witnessing ent
An important problem in quantum information processing is the certification of the dimension of quantum systems without making assumptions about the devices used to prepare and measure them, that is, in a device-independent manner. A crucial question
While the standard formulation of quantum theory assumes a fixed background causal structure, one can relax this assumption within the so-called process matrix framework. Remarkably, some processes, termed causally nonseparable, are incompatible with