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Quantum Correlations: on Their Detection, Applications, and Foundations Quantum Correlations: On their Detection, Applications, and Foundations INAUGURAL-DISSERTATION submitted in partial fulfillment of the requirements for the degree of Dr. rer. nat in the Faculty of Mathematics and Natural Sciences at the Heinrich-Heine Universität Düsseldorf presented by Jochen Szangolies from Hachenburg Düsseldorf, October 4, 2016 from the Institute for Theoretical Physics III at the Heinrich-Heine Universität Düsseldorf printed by permission of the Faculty of Mathematics and Natural Sciences at the Heinrich-Heine Universität Düsseldorf Supervisor: Prof. Dr. Dagmar Bruß Co-Supervisor: PD Dr. Hermann Kampermann Date of the oral examination: iii Declaration of Authorship Ich versichere an Eides Statt, dass die Dissertation von mir selbständig und ohne unzulässige fremde Hilfe unter Beachtung der “Grundsätze zur Sicherung guter wissenschaftlicher Praxis an der Heinrich-Heine-Universität Düsseldorf" erstellt worden ist. Signed: Date: v “The aim of science is not to open the door to infinite wisdom, but to set a limit to infinite error.” Bertolt Brecht, The Life of Galileo (1939, 1994), scene 9, 74. vii Abstract In quantum mechanics, systems may exhibit correlations that go beyond those possible in classical theories. The classical correlations form a con- vex polytope uniquely characterized by finitely many vertices, such that every classically achievable distribution of measurement probabilities can be written as a convex combination of these vertices. Consequently, any correlation that cannot be decomposed in this way is incompatible with a classical statistical theory. Such incompatibilities manifest themselves in different ways. One ex- ample is the Kochen-Specker theorem, which asserts that quantum corre- lations, in general, cannot be understood as correlations between hidden parameters whose values are independent of other, simultaneously per- formed measurements. The experimental verification of quantum mechan- ical predictions in this case is faced with the so-called problem of compat- ibility: in general, real measurements are never perfectly compatible, and thus, the assumptions underlying the Kochen-Specker theorem cannot be straightforwardly implemented. To address this issue, we present a formulation of the theorem, combin- ing it with ideas behind Leggett-Garg inequalities, that is well-defined even for imperfectly compatible observables, and which reduces to the usual for- mulation in the limit of perfect compatibility. Another important aspect of quantum correlations is the phenomenon of entanglement. Many methods to detect the entanglement of arbitrary quantum states have to be specifically taylored to that state, or else, quickly become infeasibly resource intensive. We present a novel method to detect any given state’s entanglement content by performing a sequence of ran- dom measurements on different subsystems, and constructing appropriate witness operators from these measurement by semidefinite programming. We furthermore show that this method scales favorably as compared to other methods, such as quantum state tomography. Quantum correlations can be used as a resource to perform certain tasks not classically feasible, or indeed, impossible. The third main result of this thesis is to present a novel such task: the certification of lower bounds to detector efficiencies in a device-independent scenario, where neither the quantum state nor the characterization of the measurement devices is known. To do so, we develop a method to derive Bell inequalities given only the observed measurement data, such that the violation of these in- equalities allows us to derive a minimum efficiency that the detectors must exceed in order to produce this violation. Furthermore, we discuss ap- plications of this method to (device-independent) entanglement detection, nonlocality certification without a shared reference frame, and quantum key distribution. Finally, we outline a program to recast quantum theory as a so-called principle theory, whose empirical content derives from (ideally) intuitive physical postulates. We identify the notion of an epistemic restriction, that viii is, a restriction on the amount of information that can be gathered about a system, as a possible foundation for this program. We then discuss how such an epistemic restriction emerges via logical constraints on the pre- dictability of measurement outcomes due to considerations of consistency. ix Zusammenfassung Systeme in der Quantenmechanik können Korrelationen aufweisen, die über die in klassischen Theorien möglichen hinaus gehen. Die klassischen Korrelationen bilden einen konvexen Polytopen, der durch Angabe von endlich vielen Vertices eindeutig charakterisiert werden kann, so dass jede klassisch mögliche Verteilung von Messergebnissen als konvexe Summe dieser Vertices geschrieben werden kann. Jede Korrelation die nicht in solcher Weise darstellbar ist, ist somit nicht mit einer klassischen statis- tischen Theorie vereinbar. Derartige Unvereinbarkeiten manifestieren sich in verschiedener Weise. Ein Beispiel ist das Kochen-Specker Theorem, welches aussagt, dass Quan- tenkorrelationen im Allgemeinen nicht als Korrelationen versteckter Pa- rameter, deren Wert unabhängig von anderen, gleichzeitig durchgefhrten¨ Messungen ist, verstanden werden können. Die experimentelle Überprüf- barkeit der quantenmechanischen Vorhersagen in diesem Fall sieht sich mit dem sogenannten Kompatibilitätsproblem konfrontiert: im Allgemeinen sind real durchgeführte Messungen niemals absolut kompatibel, und da- her können die Annahmen des Kochen-Specker Theorems nicht direkt in den experimentellen Kontext übersetzt werden. Um dieses Problem zu lösen, schlagen wir eine Formulierung des The- orems vor, indem wir einige Ideen, die den Leggett-Garg Ungleichungen zugrunde liegen, hinzuziehen, welche auch für nicht perfekt kompatible Observablen wohldefiniert ist, und welche für den Fall perfekter Kompati- bilität auf die ursprüngliche Formulierung reduziert. Ein weiterer wichtiger Aspekt der Quantenkorrelationen ist das Phäno- men der Verschränkung. Viele Methoden zur Detektion der Verschränkung beliebiger Quantenzustände müssen spezifisch auf diese Zustände abges- timmt sein, oder benötigen andernfalls unerreichbare Resourcen. Wir de- monstrieren eine neue Methode zur Detektion des Verschränkungsinhaltes beliebiger Zustände, indem eine Abfolge zufälliger Messungen an verschie- denen Untersystemen durchgeführt wird, woraus dann ein semidefinites Programm einen geeigneten Verschränkungszeugen konstruiert. Weiterhin zeigen wir, dass diese Methode im Vergleich mit Methoden wie etwa Quan- tenzustandstomographie ein besseres Skalierungsverhalten an den Tag legt. Quantenkorrelationen können als Resourcen für Aufgaben, die klas- sisch praktisch undurchführbar oder sogar unmöglich sind, dienen. Der dritte Beitrag dieser Dissertation legt eine neuartige Aufgabe aus diesem Bereich dar: die Zertifizierung unterer Schranken an Detektoreffizienzen im geräteunabhängigen Szenario, in welchem weder der Quantenzustand noch die Charakterisierung der Messapparatur bekannt ist. Dafür entwick- eln wir eine Methode, um Bellungleichungen lediglich aus den Messdaten zu konstruieren, so dass die Verletzung dieser Ungleichungen es uns er- laubt, die minimale Effizienz, welche die Detektoren überschreiten müssen um diese Verletzung hervorzubringen, abzuleiten. Weiterhin diskutieren wir Anwendungen dieser Methode auf die geräteunabhängige Detektion x von Verschränkung, die Feststellung nichtlokaler Korrelationen ohne ge- meinsames Bezugssystem, und die Quantenschlüsselverteilung. Schließlich beschreiben wir ein Programm, die Quantenmechanik in die Form einer Prinzipientheorie zu bringen, deren empirischer Inhalt von (ide- alerweise) intuitiv einsichtigen physikalischen Postulaten ableitbar ist. Wir identifizieren den Begriff der epistemischen Einschränkung, bei der es sich um eine Einschränkung der über ein System verfügbaren Information han- delt, als mögliche Grundlage für dieses Programm. Wir geben an, wie solch eine epistemische Einschränkung aus logischen Bedingungen bezüglich der Vorhersagbarkeit von Messergebnissen aus Konsistenzüberlegungen folgt. xi Acknowledgements The printed end result of a large project (such as, for example, the writ- ing of a doctoral thesis) often starts out with a fat lie: right there, on the cover, below the title, stands only a single name. Nothing could be further from the truth. I have benefited greatly from the kindness of heart and the generosity of spirit of those who accompanied me on my journey to this point. In com- piling this thesis and the results presented therein, my first and foremost gratitude lies with Prof. Dr. Dagmar Bruß, my advisor, whose guidance and experience proved invaluable, and whose patience and resolve were in- strumental in shaping the unique experience that working within her group was. I will always be thankful for being given this opportunity, and I will cherish these memories for years to come. No less thanks are due to PD Dr. Hermann Kampermann, whom I could always count upon to help give my vague notions a more precise and sharp formulation. I am grateful to have been able to rely on his great insight and knowledge, and derive from it many new approaches and ideas. I also wish to express heartfelt thanks to Prof. Dr. Otfried Gühne, whose encouragement got me started down this path, and likewise Dr. Matthias Kleinmann, for support and many fruitful discussions. I have been fortunate to share this journey with some of the best fellow travelers
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