Jędrzej Kaniewski

In this work, we perform a comprehensive study of the machine learning (ML) methods for the purpose of characterising the quantum set of correlations. As our main focus is on assessing the usefulness and effectiveness of the ML approach, we focus exclusively on the CHSH scenario, both the 4-dimensional variant, for which an analytical solution is known, and the 8-dimensional variant, for which no analytical solution is known, but numerical approaches are relatively well understood. We consider a wide selection of approaches, ranging from simple data science models to dense neural networks. The two classes of models that perform well are support vector machines and dense neural networks, and they are the main focus of this work. We conclude that while it is relatively easy to achieve good performance on average, it is hard to train a model that performs well on the "hard" cases, i.e., points in the vicinity of the boundary of the quantum set. Sadly, these are precisely the cases which are interesting from the academic point of view. In order to improve performance on hard cases one must, especially for the 8-dimensional problem, resort to a tailored choice of training data, which means that we are implicitly feeding our intuition and biases into the model. We feel that this is an important and often overlooked aspect of applying ML models to academic problems, where data generation or data selection is performed according to some implicit subjective criteria. In this way, it is possible to unconsciously steer our model, so that it exhibits features that we are interested in seeing. Hence, special care must be taken while determining whether ML methods can be considered objective and unbiased in the context of academic problems.

Mutually unbiased bases (MUBs) are highly symmetric bases on complex Hilbert spaces, and the corresponding rank-1 projective measurements are ubiquitous in quantum information theory. In this work, we study a recently introduced generalization of MUBs called mutually unbiased measurements (MUMs). These measurements inherit the essential property of complementarity from MUBs, but the Hilbert space dimension is no longer required to match the number of outcomes. This operational complementarity property renders MUMs highly useful for device-independent quantum information processing. It has been shown that MUMs are strictly more general than MUBs. In this work we provide a complete proof of the characterization of MUMs that are direct sums of MUBs. We then proceed to construct new examples of MUMs that are not direct sums of MUBs. A crucial technical tool for these construction is a correspondence with quaternionic Hadamard matrices, which allows us to map known examples of such matrices to MUMs that are not direct sums of MUBs. Furthermore, we show that -- in stark contrast with MUBs -- the number of MUMs for a fixed outcome number is unbounded. Next, we focus on the use of MUMs in quantum communication. We demonstrate how any pair of MUMs with d outcomes defines a d-dimensional superdense coding protocol. Using MUMs that are not direct sums of MUBs, we disprove a recent conjecture due to Nayak and Yuen on the rigidity of superdense coding for infinitely many dimensions. The superdense coding protocols arising in the refutation reveal how shared entanglement may be used in a manner heretofore unknown.

Quantum mechanics may revolutionize many aspects of modern information processing as it promises significant advantages in several fields such as cryptography, computing, and metrology. Quantum cryptography, for instance, allows us to implement protocols which are device independent, i.e., they can be proven secure under fewer assumptions. These protocols rely on using devices producing nonlocal statistics and ideally these statistics would correspond to extremal points of the quantum set in the probability space. However, even in the Clauser-Horne-Shimony-Holt (CHSH) scenario (the simplest nontrivial Bell scenario) we do not have a full understanding of the extremal quantum points. In fact, there are only a couple of analytic families of such points. Our first contribution is to introduce two families of analytical quantum extremal points by providing solutions to two families of Bell functionals. In the second part we focus on developing an analytical criteria for extremality in the CHSH scenario. A well-known Tsirelson-Landau-Masanes criterion only applies to points with uniform marginals, but a generalization has been suggested in a sequence of works by Ishizaka. We combine these conditions into a standalone conjecture, explore their technical details, and discuss their suitability. Based on the understanding acquired, we propose a set of conditions with an elegant mathematical form and an intuitive physical interpretation. Finally, we verify that both sets of conditions give correct predictions on these families of quantum extremal points.

Bell nonlocality represents one of the most striking departures of quantum mechanics from classical physics. It shows that correlations between spacelike separated systems allowed by quantum mechanics are stronger than those present in any classical theory. In a recent work [A. Tavakoli, M. Farkas, D. Rosset, J.-D. Bancal, and J. Kaniewski, Sci. Adv. 7, eabc3847 (2021)], a family of Bell functionals tailored to mutually unbiased bases (MUBs) was proposed. For these functionals, the maximal quantum violation is achieved if the two measurements performed by one of the parties are constructed out of MUBs of a fixed dimension. Here, we generalize this construction to an arbitrary incompatible pair of rank-one projective measurements. By constructing a new family of Bell functionals, we show that for any such pair there exists a Bell inequality that is maximally violated by this pair. Moreover, when investigating the robustness of these violations to noise, we demonstrate that the realization which is most robust to noise is not generated by MUBs.

Bell non-locality represents one of the most striking departures of quantum mechanics from classical physics. It shows that correlations between space-like separated systems allowed by quantum mechanics are stronger than those present in any classical theory. In a recent work [Sci. Adv. 7, eabc3847 (2021)], a family of Bell functionals tailored to mutually unbiased bases (MUBs) is proposed. For these functionals, the maximal quantum violation is achieved if the two measurements performed by one of the parties are constructed out of MUBs of a fixed dimension. Here, we generalize this construction to an arbitrary incompatible pair of rank-one projective measurements. By constructing a new family of Bell functionals, we show that for any such pair there exists a Bell inequality that is maximally violated by this pair. Moreover, when investigating the robustness of these violations to noise, we demonstrate that the realization which is most robust to noise is not generated by MUBs.

The existence of incompatible measurements, i.e. measurements which cannot be performed simultaneously on a single copy of a quantum state, constitutes an important distinction between quantum mechanics and classical theories. While incompatibility might at first glance seem like an obstacle, it turns to be a necessary ingredient to achieve the so-called quantum advantage in various operational tasks like random access codes or key distribution. To improve our understanding of how to quantify incompatibility of quantum measurements, we define and explore a family of incompatibility measures based on non-commutativity. We investigate some basic properties of these measures, we show that they satisfy some natural information-processing requirements and we fully characterize the pairs which achieve the highest incompatibility (in a fixed dimension). We also consider the behavior of our measures under different types of compositions. Finally, to link our new measures to existing results, we relate them to a robustness-based incompatibility measure and two operational scenarios: random access codes and entropic uncertainty relations.