This thesis studies different problems in quantum information theory and the foundations of quantum mechanics. These include the quantum marginal problem, the problem of causal inference in quantum mechanics, and the problem of indefinite causal order in quantum processes.
We start by considering an instance of the quantum marginal problem in which our goal is to detect genuine multiparticle entanglement from the marginal information, i.e. correlations in the subsystems. In simple words, genuine multiparticle entanglement means that all particles are entangled with each other. Moreover, we consider an exotic case where the marginals themselves are separable, i.e. do not manifest entanglement if considered separately. Our results show that this phenomenon, which we call emergence of multiparticle entanglement, occurs frequently and for an arbitrary number of particles. In particular, we present a systematic method to look for such states and present various examples of systems up to 6 qubits (two-level systems). Interestingly, already for four qubits there exist a pure state with this properties which suggests that this phenomenon can be observed in the experiment. In the subsequent part of the thesis we define and study a particular class of genuine entangled states, called hypergraph states, in systems of qudits (d-level systems). This class of states is a generalization of graph states, which are used in measurement-based quantum computing and error-correcting codes. Hypergraph states can be obtained by applying certain sequence of entangling gates, associated with hyperedges, on systems of qudits, associated with vertices. In this thesis we provide a detailed analysis of equivalence of tripartite hypergraph states in dimension 3 and 4 under local operations.
Then we pass on to the problem of explaining correlations observed in experiment by classical causal models. A particular example of a causal model is a local hidden variable model of Bell’s test. Cause-effect relations, or causal links, in causal models are given by the underlying causal structures, which often can be represented by graphs. Given a causal structure one can derive constraints for correlations to be compatible with this structure, which in the case of Bell’s theorem are the famous Bell inequalities. Alternatively, given experimental data, the task would be to determine the underlined causal model, which is a problem of causal inference. In some experiments of causal inference the correlations among all variables cannot be accessed or are not collected. In this case one faces a type of the marginal problem where one has to judge about possible underlined causal structures from marginal data. Clearly, the success of the causal inference in this case depends strongly on the configuration of accessible marginals, which is known as the marginal scenario. In this thesis we provide a general theory connecting marginal scenario and possible causal structures. We derive a necessary condition on causal structures to be distinguishable from a given marginal scenario. Among others, this result can help us to find new interesting scenarios for nonlocality tests.
Finally, we discuss the problem of indefinite causal order in quantum mechanics. Causal order puts restrictions on causal relations in two systems of random variables generated by two different events. In particular, it restricts these causal links to be directed in the same way, from one event to the other. An example of causal order is a space-time manifold. Recently, it has been realized that physical theories do not necessarily have to comply with the idea of a definite causal order. For example, one can imagine a theory where the causal order is a dynamical element of this theory and can be in a sort of “quantum superposition". In this thesis we derive inequalities, similar to those of Bell, but for testing indefiniteness of causal order. In particular, inequalities are derived for information-theoretic quantities allowing for testing information flow in processes with indefinite causal order.