Local realism is the worldview in which physical properties of objects exist independently of measurement and where physical influences cannot travel faster than the speed of light. Bell’s theorem states that this worldview is incompatible with the predictions of quantum mechanics, as is expressed in Bell’s inequalities. Previous experiments convincingly supported the quantum predictions. Yet, every experiment requires assumptions that provide loopholes for a local realist explanation. Here, we report a Bell test that closes the most significant of these loopholes simultaneously. Using a well-optimized source of entangled photons, rapid setting generation, and highly efficient superconducting detectors, we observe a violation of a Bell inequality with high statistical significance. The purely statistical probability of our results to occur under local realism does not exceed $3.74\times {10}^{-31}$, corresponding to an 11.5 standard deviation effect.

A central result in the foundations of quantum mechanics is the Kochen-Specker theorem. In short, it states that quantum mechanics is in conflict with classical models in which the result of a measurement does not depend on which other compatible measurements are jointly performed. Here compatible measurements are those that can be implemented simultaneously or, more generally, those that are jointly measurable. This conflict is generically called quantum contextuality. In this review, an introduction to this subject and its current status is presented. Several proofs of the Kochen-Specker theorem and different notions of contextuality are reviewed. How to experimentally test some of these notions is explained, and connections between contextuality and nonlocality or graph theory are discussed. Finally, some applications of contextuality in quantum information processing are reviewed.

The notion of (non)contextuality pertains to sets of properties measured one subset (context) at a time. We extend this notion to include so-called inconsistently connected systems, in which the measurements of a given property in different contexts may have different distributions, due to contextual biases in experimental design or physical interactions (signaling): a system of measurements has a maximally noncontextual description if they can be imposed a joint distribution on in which the measurements of any one property in different contexts are equal to each other with the maximal probability allowed by their different distributions. We derive necessary and sufficient conditions for the existence of such a description in a broad class of systems including Klyachko-Can-Binicioğlu-Shumvosky-type (KCBS), EPR-Bell-type, and Leggett-Garg-type systems. Because these conditions allow for inconsistent connectedness, they are applicable to real experiments. We illustrate this by analyzing an experiment by Lapkiewicz and colleagues aimed at testing contextuality in a KCBS-type system.

We report on a search for mutually unbiased bases (MUBs) in six dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a detailed map of where in the landscape the MUB triplets are situated. We use available tools, such as the theory of the discrete Fourier transform, to organize our results. Finally, we present some evidence for the conjecture that there exists a four dimensional family of complex Hadamard matrices of order 6. If this conjecture is true the landscape in which one may search for MUBs is much larger than previously thought.