Characterizing finite-dimensional quantum behavior

We study and extend the semidefinite programming (SDP) hierarchies introduced in [Phys. Rev. Lett. 115, 020501] for the characterization of the statistical correlations arising from finite dimensional quantum systems. First, we introduce the dimension-constrained noncommutative polynomial optimization (NPO) paradigm, where a number of polynomial inequalities are defined and optimization is conducted over all feasible operator representations of bounded dimensionality. Important problems in device independent and semi-device independent quantum information science can be formulated (or almost formulated) in this framework. We present effective SDP hierarchies to attack the general dimension-constrained NPO problem (and related ones) and prove their asymptotic convergence. To illustrate the power of these relaxations, we use them to derive new dimension witnesses for temporal and Bell-type correlation scenarios, and also to bound the probability of success of quantum random access codes.Comment: 17 page

The simplest causal inequalities and their violation

In a scenario where two parties share, act on and exchange some physical resource, the assumption that the parties' actions are ordered according to a definite causal structure yields constraints on the possible correlations that can be established. We show that the set of correlations that are compatible with a definite causal order forms a polytope, whose facets define causal inequalities. We fully characterize this causal polytope in the simplest case of bipartite correlations with binary inputs and outputs. We find two families of nonequivalent causal inequalities; both can be violated in the recently introduced framework of process matrices, which extends the standard quantum formalism by relaxing the implicit assumption of a fixed causal structure. Our work paves the way to a more systematic investigation of causal inequalities in a theory-independent way, and of their violation within the framework of process matrices.Comment: 7 + 4 pages, 2 figure

Witnessing causal nonseparability

Our common understanding of the physical world deeply relies on the notion that events are ordered with respect to some time parameter, with past events serving as causes for future ones. Nonetheless, it was recently found that it is possible to formulate quantum mechanics without any reference to a global time or causal structure. The resulting framework includes new kinds of quantum resources that allow performing tasks - in particular, the violation of causal inequalities - which are impossible for events ordered according to a global causal order. However, no physical implementation of such resources is known. Here we show that a recently demonstrated resource for quantum computation - the quantum switch - is a genuine example of "indefinite causal order". We do this by introducing a new tool - the causal witness - which can detect the causal nonseparability of any quantum resource that is incompatible with a definite causal order. We show however that the quantum switch does not violate any causal nequality.Comment: 15 + 12 pages, 5 figures. Published versio

Higher order interference theories

In dieser Masterarbeit wurden Theorien untersucht, die neben Interferenz zweiter Ordnung (wie in der Quantenmechanik, wo diese Interferenz durch Einträge einer Dichtematrix aufgefasst werden können) auch Interferenzterme höherer Ordnung erlauben. Die Motivation dafür lieferte die Theorie der density cubes, die von Dakic, Paterek und Brukner ausgearbeitet wurde und es erlaubt, Interferenz dritter Ordnung zu beschreiben. Innerhalb der density cube Theorie sind Transformationen möglich, die in der Quantenmechanik verboten wären. Eine Beispiel für eine solche Transformation (T) erlaubt es, die quantenmechanische Grenze (Tsirelson-Schranke) für das zeitliche Äquivalent der CHSH-Ungleichung zu verletzen. Es wird gezeigt, dass die density cube Theorie und allgemein alle Theorien, die zusätzlich zur Interferenz zweiter Ordnung noch Interferenz n-ter Ordnung ermöglichen, nicht als Verallgemeinerung der Quantenmechanik fungieren können, weil sie nicht zugleich (1) die Quantenmechanik als Spezialfall erklären und (2) einen von ihr unterschiedlichen empirischen Gehalt haben können. Außerdem wird das Verhältnis zwischen Verletzung der zeitlichen Tsirelson-Schranke (im CHSH-Szenario) und der Menge an Information, die maximal zwischen zwei Zeitpunkten ausgetauscht wird, untersucht und eine neue Transformation, welche die Tsirelson-Schranke maximal verletzt, präsentiert.The aim of this thesis is to examine certain theories that are more general than quantum mechanics. Quantum mechanics itself can be understood to describe interference of order 2, as described by density matrices. Here we consider those generalizations of quantum mechanics that allow for higher order interference terms. The motivation for this assessment was given by the density cube theory, developed by Dakic, Paterek and Brukner, which accounts for interference of third order. I will show that the density cube theory, as well as any other higher order interference theory which adds interference of order n (and only n) to quantum interference cannot achieve the following two natural requirements for a generalization of quantum mechanics: (1) recovering quantum mechanics as a special case and (2) predicting experimental results differing from it. I is shown that the transformation T for density cubes is (considering that composite systems are obtained using the tensor product) not completely positive, and hence not physically realisable. Then the general theorem of incompatibility between higher order interference theories of arbitrary order n and quantum mechanics will be proven: no theory adding the possibility for interference of order n (and only n) can both satisfy conditions (1) and (2) from above. Finally, the connection between density cube theory and the strength of signalling is discussed, and a modification of T is presented, which allows maximal violation of the quantum bound