A new structure for difference matrices over abelian pp-groups

A difference matrix over a group is a discrete structure that is intimately related to many other combinatorial designs, including mutually orthogonal Latin squares, orthogonal arrays, and transversal designs. Interest in constructing difference matrices over $2$-groups has been renewed by the recent discovery that these matrices can be used to construct large linking systems of difference sets, which in turn provide examples of systems of linked symmetric designs and association schemes. We survey the main constructive and nonexistence results for difference matrices, beginning with a classical construction based on the properties of a finite field. We then introduce the concept of a contracted difference matrix, which generates a much larger difference matrix. We show that several of the main constructive results for difference matrices over abelian $p$-groups can be substantially simplified and extended using contracted difference matrices. In particular, we obtain new linking systems of difference sets of size $7$ in infinite families of abelian $2$-groups, whereas previously the largest known size was $3$.Comment: 27 pages. Discussion of new reference [LT04

Non-Independent Components Analysis

A seminal result in the ICA literature states that for $AY = \varepsilon$, if the components of $\varepsilon$ are independent and at most one is Gaussian, then $A$ is identified up to sign and permutation of its rows (Comon, 1994). In this paper we study to which extent the independence assumption can be relaxed by replacing it with restrictions on higher order moment or cumulant tensors of $\varepsilon$. We document new conditions that establish identification for several non-independent component models, e.g. common variance models, and propose efficient estimation methods based on the identification results. We show that in situations where independence cannot be assumed the efficiency gains can be significant relative to methods that rely on independence

New Concepts in Quantum Metrology: Dynamics, Machine Learning, and Bounds on Measurement Precision

Diese kumulative Promotionsarbeit befasst sich mit theoretischer Quantenmetrologie, der Theorie von Messung und Schätzung unter Zuhilfenahme von Quantenressourcen. Viele Vorschläge für quantenverbesserte Sensoren beruhen auf der Präparation von nichtklassischen Anfangszuständen und integrabler Dynamik. Allerdings sind solche nichtklassischen Zustände schwierig zu präparieren und gegen Dekohärenz zu schützen. Alternativ schlagen wir in dieser Promotionsarbeit sogenannte quantenchaotische Sensoren vor, die auf klassischen Anfangszuständen beruhen, die einfach zu präparieren sind, wobei Quantenverbesserungen an der Dynamik vorgenommen werden. Diese Herangehensweise hat ihren Ursprung darin, dass sowohl Quantenchaos als auch Quantenmetrologie über die Empfindlichkeit für kleine Änderungen in der Dynamik charakterisiert werden. Wir erforschen unterschiedliche Arten von Dynamik am Beispiel des Modells eines gestoßenen Quantenkreisels ("kicked top"), dessen Dynamik durch nichtlineare Kontrollpulse quantenchaotisch wird. Außerdem zeigen wir, dass Quantenchaos in der Lage ist, schädlichen Dekohärenzeffekte abzuschwächen. Insbesondere präsentieren wir einen Vorschlag für ein quantenchaotisches Cäsiumdampf-Magnetometer. Mit der Hilfe von Bestärkendem Lernen verbessern wir Zeitpunkt und Stärke der nichtlinearen Pulse im Modell des gestoßenen Quantenkreisels mit Superradianzdämpfung. Für diesen Fall finden wir, dass die Kontrollstrategie als eine dynamische Form der Spin-Quetschung verstanden werden kann. Ein anderer Teil dieser Promotionsarbeit beschäftigt sich mit bayesscher Quantenschätzung und insbesondere mit dem Problem der heuristischen Gestaltung von Experimenten. Wir trainieren neuronale Netze mit einer Kombination aus überwachtem und bestärkendem Lernen, um diese zu schnellen und starken Heuristiken für die Gestaltung von Experimenten zu machen. Die Vielseitigkeit unserer Methode zeigen wir anhand von Beispielen zu Einzel- und Mehrparameterschätzung, in denen die trainierten neuronalen Netze die Leistung der modernsten Heuristiken übertreffen. Außerdem beschäftigen wir uns mit einer lange unbewiesenen Vermutung aus dem Bereich der Quantenmetrologie: Wir liefern einen Beweis für diese Vermutung und finden einen Ausdruck für die maximale Quantenfischerinformation für beliebige gemischte Zustände und beliebige unitäre Dynamik, finden Bedingungen für optimale Zustandspräparation und optimale dynamische Kontrolle, und verwenden diese Ergebnisse, um zu beweisen, dass die Heisenberg-Schranke sogar mit thermischen Zuständen beliebiger (endlicher) Temperatur erreicht werden kann.This cumulative thesis is concerned with theoretical quantum metrology, the theory of measurement and estimation using quantum resources. Possible applications of quantum-enhanced sensors include the measurement of magnetic fields, gravitational wave detection, navigation, remote sensing, or the improvement of frequency standards. Many proposals for quantum-enhanced sensors rely on the preparation of non-classical initial states and integrable dynamics. However, such non-classical states are generally difficult to prepare and to protect against decoherence. As an alternative, in this thesis, we propose so-called quantum-chaotic sensors which make use of classical initial states that are easy to prepare while quantum enhancements are applied to the dynamics. This approach is motivated by the insight that quantum chaos and quantum metrology are both characterized by the sensitivity to small changes of the dynamics. At the example of the quantum kicked top model, where nonlinear control pulses render the dynamics quantum-chaotic, we explore different dynamical regimes for quantum sensors. Further, we demonstrate that quantum chaos is able to alleviate the detrimental effects of decoherence. In particular, we present a proposal for a quantum-chaotic cesium-vapor magnetometer. With the help of reinforcement learning, we further optimize timing and strength of the nonlinear control pulses for the kicked top model with superradiant damping. In this case, the optimized control policy is identified as a dynamical form of spin squeezing. Another part of this thesis deals with Bayesian quantum estimation and, in particular, with the problem of experiment design heuristics. We train neural networks with a combination of supervised and reinforcement learning to become fast and strong experiment design heuristics. We demonstrate the versatility of this method using examples of single and multi-parameter estimation where the trained neural networks surpass the performance of well-established heuristics. Finally, this thesis deals with a long-time outstanding conjecture in quantum metrology: we prove this conjecture and find an expression for the maximal quantum Fisher information for any mixed initial state and any unitary dynamics, provide conditions for optimal state preparation and optimal control of the dynamics, and utilize these results to prove that Heisenberg scaling can be achieved even with thermal states of arbitrary (finite) temperature

Inference in Networking Systems with Designed Measurements

Networking systems consist of network infrastructures and the end-hosts have been essential in supporting our daily communication, delivering huge amount of content and large number of services, and providing large scale distributed computing. To monitor and optimize the performance of such networking systems, or to provide flexible functionalities for the applications running on top of them, it is important to know the internal metrics of the networking systems such as link loss rates or path delays. The internal metrics are often not directly available due to the scale and complexity of the networking systems. This motivates the techniques of inference on internal metrics through available measurements. In this thesis, I investigate inference methods on networking systems from multiple aspects. In the context of mapping users to servers in content delivery networks, we show that letting user select a server that provides good performance from a set of servers that are randomly allocated to the user can lead to optimal server allocation, of which a key element is to infer the work load on the servers using the performance feedback. For network tomography, where the objective is to estimate link metrics (loss rate, delay, etc.) using end-to-end measurements, we show that the information of each end-to-end measurement can be quantified by Fisher Information and the estimation error of link metrics can be efficiently reduced if the allocation of measurements on paths is designed to maximize the overall information. Last but not least, in the context of finding the most reliable path for routing from a source to a destination in a network while minimizing the cost of exploring lossy paths, the trade-off between exploiting the best paths based on estimated loss rates and taking the risk to explore worse paths to improve the estimation is investigated, and online learning methods are developed and analyzed. The performance of the developed techniques are evaluated with simulations