Information causality as a tool for bounding the set of quantum correlations

Information causality was initially proposed as a physical principle aimed at deriving the predictions of quantum mechanics on the type of correlations observed in the Bell experiment. In the same work, information causality was famously shown to imply the Uffink inequality that approximates the set of quantum correlations and rederives Tsirelson's bound of the Clauser-Horne-Shimony-Holt inequality. This result found limited generalizations due to the difficulty of deducing implications of the information causality principle on the set of nonlocal correlations. In this paper, we present a simple technique for obtaining polynomial inequalities from information causality, bounding the set of physical correlations in any Bell scenario. To demonstrate our method, we derive a family of inequalities which non-trivially constrains the set of nonlocal correlations in Bell scenarios with binary outcomes and equal number of measurement settings. Finally, we propose an improved statement of the information causality principle, obtain tighter constraints for the simplest Bell scenario that goes beyond the Uffink inequality, and recovers a part of the boundary of the quantum set.Comment: 4 pages + Appendix, 2 figures, comments are welcom

Quanten-Hypergraph-Zustände und die Theorie der Mehrteilchen-Verschränkung

This thesis is devoted to learning different aspects of quantum entanglement theory. More precisely, it concerns a characterization of certain classes of pure multipartite entangled states, their nonlocal and entanglement properties, comparisons with the other well-studied classes of states and, finally, their utilization in certain quantum information processing tasks. The most extensive part of the thesis explores an interesting class of pure multipartite entangled states, quantum hypergraph states. These states are generalizations of the renowned class of graph states. Here we cover their nonlocal properties in various scenarios, derive graphical rules for unitary transformations and Pauli bases measurements. Using these rules, we characterize entanglement classes of hypergraph states under local operations, obtain tight entanglement witnesses, and calculate entanglement measures for hypergraph states. Finally, we apply all the aforementioned analysis to endorse hypergraph states as powerful resource states for measurement-based quantum computation and quantum error-correction. The rest of the thesis is devoted to three disjoint problems, but all of them are still in the scope of entanglement theory. First, using mathematical structure of linear matrix pencils, we coarse grain entanglement in tripartite pure states of local dimensions 2 x m x n under the most general local transformations. In addition, we identify the structure of generic states for every m and n and see that for certain dimensions there is a resemblance between bipartite and tripartite entanglement. Second, we consider the following question: Can entanglement detection be improved, if in addition to the expectation value of the measured witness, we have knowledge of the expectation value of another observable? For low dimensions we give necessary and sufficient criterion that such two product observables must satisfy in order to be able to detect entanglement. Finally, we derive a general statement that any genuine N-partite entangled state can always be projected on any of its k-partite subsystems in a way that the new state in genuine k-partite entangled.Diese Arbeit ist verschiedenen Aspekten der Verschränkungstheorie gewidmet. Genauer gesagt, beschäftigt sie sich mit der Charakterisierung verschiedener Klassen reiner mehrteilchenverschränkter Zustände, sowie ihrer nicht-lokalen und Verschränkungseigenschaften, Vergleichen mit anderen bekannten Klassen von Zuständen und, letztendlich, ihrer Verwendung in der Quanteninformationsverarbeitung. Der größte Teil dieser Arbeit beschäftigt sich mit einer interessanten Klasse von reinen mehrteilchenverschränkten Zuständen, den Hypergraphen Zuständen. Diese Zustände bilden eine Verallgemeinerung der weithin bekannten Graphen Zustände. Wir werden ihre nicht-lokalen Eigenschaften untersuchen und graphische Regeln für ihr Verhalten unter unitären Transformationen sowie Messungen in der Pauli Basis herleiten. Unter Verwendung dieser Regeln werden Verschränkungsklassen unter lokalen Operationen charakterisiert, und optimale Verschränkungszeugen sowie Verschränkungsmaße berechnet. Zuletzt wird, unter Berücksichtigung der vorangegangenen Analyse, gezeigt, dass Hypergraphen Zustände eine Ressource für Messungsbasierte Quantencomputer und Quantenfehlerkorrektur bilden. Der verbleibende Teil dieser Arbeit beschäftigt sich mit drei unterschiedlichen Problemen im Rahmen der Verschränkungstheorie. Zuerst wird die Verschränkung reiner Dreiteilchenzustände mit lokalen Dimensionen 2 x m x n unter den allgemeinsten Transformationen untersucht. Dazu wird eine spezielle mathematische Struktur verwendet, die so genannten Matrix Pencils. Zudem identifizieren wir die Struktur von generischen Zuständen für alle m und n und wir werden sehen, dass in bestimmten Dimensionen eine Ähnlichkeit zwischen Zweiteilchen- und Dreiteilchenverschränkung besteht. Danach wird der Frage nachgegangen: Kann Verschränkungsdetektion verbessert werden, wenn man zusätzlich zum Erwartungswertes des Verschränkungszeugen noch den Erwartungswert einer anderen Observablen kennt? Für niedrige Dimensionen werden notwendige und hinreichende Bedingungen hergeleitet, die zwei Produktobservablen erfüllen müssen um Verschränkung detektieren zu können. Zuletzt wird die allgemeine Aussage bewiesen, dass jeder echte N-Teilchen verschränkte Zustand immer auf seine k-Teilchen Subsysteme projiziert werden kann, sodass dieser Zustand echt k-Teilchen verschränkt ist

Quantifying Causal Influences in the Presence of a Quantum Common Cause

Quantum mechanics challenges our intuition on the cause-effect relations in nature. Some fundamental concepts, including Reichenbach's common cause principle or the notion of local realism, have to be reconsidered. Traditionally, this is witnessed by the violation of a Bell inequality. But are Bell inequalities the only signature of the incompatibility between quantum correlations and causality theory? Motivated by this question, we introduce a general framework able to estimate causal influences between two variables, without the need of interventions and irrespectively of the classical, quantum, or even postquantum nature of a common cause. In particular, by considering the simplest instrumental scenario-for which violation of Bell inequalities is not possible-we show that every pure bipartite entangled state violates the classical bounds on causal influence, thus, answering in negative to the posed question and opening a new venue to explore the role of causality within quantum theory

Causal inference with imperfect instrumental variables

Instrumental variables allow for quantification of cause and effect relationships even in the absence of interventions. To achieve this, a number of causal assumptions must be met, the most important of which is the independence assumption, which states that the instrument and any confounding factor must be independent. However, if this independence condition is not met, can we still work with imperfect instrumental variables? Imperfect instruments can manifest themselves by violations of the instrumental inequalities that constrain the set of correlations in the scenario. In this article, we establish a quantitative relationship between such violations of instrumental inequalities and the minimal amount of measurement dependence required to explain them for the case of discrete observed variables. As a result, we provide adapted inequalities that are valid in the presence of a relaxed measurement dependence assumption in the instrumental scenario. This allows for the adaptation of existing and new lower bounds on the average causal effect for instrumental scenarios with binary outcomes. Finally, we discuss our findings in the context of quantum mechanics