Geometrical aspects of qudits concerning Bell inequalities

The aim of this thesis is to investigate quantum entanglement and quantum nonlocality of bipartite finite-dimensional systems (bipartite qudits). Entanglement is one of the most fascinating non-classical features of quantum theory, and besides its impact on our view of the world, it can be exploited for applications such as quantum cryptography and quantum computing. (...) Although entanglement and nonlocality are ordinarily regarded as one and the same, under close consideration this cannot be taken for granted. The reason for this is that entanglement is defined by the mathematical structure of a quantum state in a composite Hilbert space, whereas nonlocality signifies that the statistical behaviour of a system cannot be described by a local realistic theory. For the latter it is essential that the correlation probabilities of such theories obey so-called Bell inequalities, which are violated for certain quantum states. The main focus of this thesis is on the comparison of both properties with the objective of understanding their relation. (...) Because of the fact that the correlation probabilities in general depend on the measurement settings it is necessary to optimise these in order to reveal nonlocality. This problem is solved for a particular Bell inequality (CGMLP) by means of a self-developed numerical search algorithm. The method is then applied to density matrices of a subspace spanned by the projectors of maximally entangled two-qudit states. This set of states (...) serves to visualise and analyse the state space geometrically.Comment: Diploma Thesis, University of Vienna (2008

A composite parameterization of unitary groups, density matrices and subspaces

Unitary transformations and density matrices are central objects in quantum physics and various tasks require to introduce them in a parameterized form. In the present article we present a parameterization of the unitary group $\mathcal{U}(d)$ of arbitrary dimension $d$ which is constructed in a composite way. We show explicitly how any element of $\mathcal{U}(d)$ can be composed of matrix exponential functions of generalized anti-symmetric $\sigma$-matrices and one-dimensional projectors. The specific form makes it considerably easy to identify and discard redundant parameters in several cases. In this way, redundancy-free density matrices of arbitrary rank $k$ can be formulated. Our construction can also be used to derive an orthonormal basis of any $k$-dimensional subspaces of $\mathbb{C}^d$ with the minimal number of parameters. As an example it will be shown that this feature leads to a significant reduction of parameters in the case of investigating distillability of quantum states via lower bounds of an entanglement measure (the $m$-concurrence).Comment: 13 pages, 1 figur

Experimentally implementable criteria revealing substructures of genuine multipartite entanglement

We present a general framework that reveals substructures of genuine multipartite entanglement. Via simple inequalities it is possible to discriminate different sets of multipartite qubit states. These inequalities are beneficial regarding experimental examinations as only local measurements are required. Furthermore, the number of observables scales favorably with system size. In exemplary cases we demonstrate the noise resistance and discuss implementations.Comment: 14 pages, 4 figure

Entanglement and correlations in composite quantum systems

Verschränkung ist eine der wichtigsten Besonderheiten der Quantentheorie, welche kein klassisches Analogon besitzt. Aufgrund dieser ist es nicht länger angemessen zusammengesetzte Systeme lediglich als eine Kombination von unabhängig voneinander beschreibbaren Subsystemen zu betrachten. Anstatt dessen gibt es Quantenzustände die inseparabel bezüglich der einzelnen Bestandteile des Systems sind. Wohingegen Verschränkung zunächst als eine Art künstlicher Fehler der Theorie angesehen wurde, hat sich deren Existenz mittlerweile durch zahlreiche Experimente bestätigt. Diese Gegebenheit hat weitreichende Konsequenzen. Auf einer fundamentalen Ebene lässt sich hiermit zeigen, daß es keine lokal-realistische Alternative zur Quantentheorie gibt. Zusätzlich stellt Verschränkung eine Ressource für neue Technologien der Informationsverarbeitung dar, wie z.B. Quantenkryptographie, Dense Coding, Quantenteleportation oder der Quantencomputer. Trotz intensiver Forschung in den letzten Jahrzehnten gibt es noch viele offene Fragen bezüglich Verschränkung und deren Manifestationen. Dies gilt insbesondere für Verschränkung in komplexen Vielteilchensystemen. Gegenstand dieser Dissertation ist es Verschränkung und nicht-lokal-realitsche Korrelationen in zusammengesetzten endlich-dimensionalen Quantensystemen (Multipartite Qudits) zu untersuchen und zu verstehen. Um die Analyse solcher Systeme zu erleichtern, werden neue mathematische Hilfsmittel, wie zum Beispiel praktische Parameterisierungen von unitären Gruppen, Dichtematrizen und Unterräumen vorgestellt. Die Struktur von Verschränkung in Vielteilchensystemen wird betrachtet, und eine exakte Charakterisierung von Multilevel-Vielteilchenverschränkung wird eingeführt. Es werden Methoden zur Verschränkungsdetektion angegeben, und deren experimentelle Implementierung diskutiert. Ein weiteres Thema ist die Quantifizierung von Verschränkung. In diesem Zusammenhang wird ein nützliches Maß für Vielteilchenverschränkung vorgestellt. Darüber hinaus wird auch die Klassifizierung von Vielteilchenverschränkung thematisiert, und ein systematischer Ansatz zur Unterscheidung verschiedener Klassen angegeben. Der letzte Teil dieser Arbeit beschäftigt sich mit Relationen zwischen Verschränkung und anderen fundamentalen Aspekten der Quantentheorie. Insbesondere wird eine Verbindung zwischen dem Komplementaritätsprinzip und dem Separabilitätsproblem hergestellt. Besondere Aufmerksamkeit gilt auch dem Zusammenhang zwischen Verschränkung und nicht-lokal-realitischen Korrelationen. Hier wird eine geometrische Struktur, welche diskreten zusammengesetzten Systemen zugrunde liegt, verwendet, um deren Verschiedenartigkeit veranschaulichen zu können.Entanglement is a key feature of quantum theory which has no classical analogue. Due to this feature it is no longer accurate to regard composite systems as a mere combination of independently describable subsystems. Instead there are quantum states which are inseparable with respect to the individual parts of the system. Initially regarded as an artifact of the theory, numerous experiments performed in the last decades have provided evidence of the existence of entanglement in nature. The consequences of entanglement are far-reaching. On a fundamental level, it allows to demonstrate that there is no local-realistic alternative to quantum theory. In addition, entanglement also serves as a resource for novel information processing technologies such as quantum cryptography, dense coding, quantum teleportation and quantum computing. Despite extensive research in recent years, several questions concerning entanglement and its manifestations still remain open --- especially in complex many-body systems. The aim of this dissertation is to investigate entanglement and non-local-realistic correlations in composite finite-dimensional quantum systems (i.e. multipartite qudits). In order to simplify the analysis of those systems, we present new mathematical tools such as convenient parameterizations for unitary groups, density matrices and subspaces. We study the structure of entanglement in multipartite systems and introduce a precise characterization of multilevel-multipartite entanglement. We consider methods for entanglement detection and discuss their implementation in experiments. Another problem treated in this thesis concerns the quantification of entanglement. Here, we introduce a useful measure of multipartite entanglement and derive computable lower bounds. Moreover, the classification of multipartite entanglement is also addressed, where a systematic approach for discriminating between different classes is given. The last part of this work deals with relations between entanglement and other foundational aspects of quantum theory. Specifically, we establish a link between complementarity and the separability problem. Particular attention is also devoted to the connection between entanglement and non-local-realistic correlations. Here, we exploit a geometric structure underlying discrete composite systems to illustrate their dissimilarities

Improving animal health and welfare in organic cattle milk production through breeding and management

Overall summary of main results, discussion and conclusions WP1 The aim in WP1 was to map the performance of local breeds compared to commercial dairy breeds on organic farms in Austria, Switzerland, Sweden, Poland and Germany with regard to health associated traits, fertility and production traits. In a study covering the whole population of the selected breeds, managed under organic conditions during the period 1.7.2011 to 30.6.2014, we found lower milk yields for local breeds. In general we found better fertility performance (less days open, shorter calving intervals, lower number of inseminations) for at least one of the local breeds involved in each country. We also detected lower proportions of milk records with somatic cell counts above 100.000 cells/ ml milk for in at least one of the local breeds in Austria, Switzerland and Sweden, but not so for Poland where the commercial breed performed best. Moreover, we found a lower proportion of test day records with a fat: protein ratio above 1.5 in the first 100 days in milk for many local breeds (Grey Cattle (AL) and Original Braunvieh (OB) in Switzerland. AL by trend in Austria, Swedish Red (SRB) in Sweden and Polish Red and White (ZR) in Poland) hinting at a lower risk of subclinical ketosis for some of the local breeds compared to the respective commercial breed. We could not find breed differences regarding overall occurrence of veterinary treatments or those due to fertility or leg/ claw problems in Austrian data, but found that AL had less treatments due to udder problems than the commercial Braunvieh (BV). In Sweden, the local breed SRB had less overall treatments, fewer treatments due to fertility or udder disorders and also showed less remarks on claw or leg disorders than the commercial breed. We did not find breed differences regarding treatments due to metabolic disorders in Swedish data. We analyzed Austrian and Swiss data from culled cows on productive life span and detected a superiority of local breeds for this trait, but higher lifetime production (kg ECM) in the commercial breed. The second study on cows managed under very similar conditions in Sweden (period: 1.7.2011 to 30.6.2014) and Germany (period 1.7.2011 to 30.6.2015) revealed no significant breed differences in total ECM yield in German data, while we found higher milk fat and milk protein contents for the local breed Original Red Angler Cattle breed (AAZ). In Sweden, we found the highest ECM yield in the commercial breed Swedish Holstein (SH). The local breed Swedish Polled (SKB) had the highest fat and protein contents, followed by the second local breed Swedish Red (SRB), both differed significantly from the respective commercial breed SH and also among each other. We could not detect significant breed differences in the German data set with regard to fertility traits, although the local AAZ tended to have fewer days open than the commercial breed HO. In Sweden, we did not find breed differences for calving interval, but both local breeds had fewer days open, and SKB needed fewer inseminations compared to SH. No breed differences in any of the two countries regarding the proportion of test day records with a SCC content of over 100.000 cells per ml milk was found. We assessed the risk of subclinical ketosis during the first 100 days in milk by comparing fat protein ratios (FPR), applying differing thresholds in Germany (AAZ >1.7, HO >1.5) per breed, without detecting breed differences. However, in Sweden (with a threshold of >1.5) where the local SRB performed best, SKB showed an intermedium position, while SH had the highest proportion of FPR >1.5. Occurrence of veterinary treatments and those due to fertility problems did not differ between German breeds. Data on treatment due to metabolic disorders and leg or claw problems had low incidences, but models did not converge. In Sweden, we found the lowest incidence of overall veterinary treatments and treatments due to mastitis in the local breed SRB, but the local breed SKB did not differ from the commercial SH in neither of the two traits. We did not find breed differences regarding veterinary treatments due to fertility problems or diagnosis of claw/leg problems during claw trimming in the Swedish data set. Incidences of treatments due to metabolic disorders were low in all Swedish breeds, but models for this trait failed to converge. From our finding’s we conclude that local breeds partly have advantages with regard to many functional traits studied and that their potential should be better exploited in future, especially on farms with a medium production level (most clearly shown in the small study on German and Swedish farms). We also found overall high proportions of test day records with elevated SCC in Germany and even more so in Poland from which we deduct a great potential for further overall management improvement regardless of the involved breeds in these countries

Measure of genuine multipartite entanglement with computable lower bounds

We introduce an intuitive measure of genuine multipartite entanglement which is based on the well-known concurrence. We show how lower bounds on this measure can be derived that also meet important characteristics of an entanglement measure. These lower bounds are experimentally implementable in a feasible way enabling quantification of multipartite entanglement in a broad variety of cases.Comment: 5 pages, 2 figure

Glycolytic Shunts Replenish the Calvin-Benson-Bassham Cycle as Anaplerotic Reactions in Cyanobacteria

The recent discovery of the Entner-Doudoroff (ED) pathway as a third glycolytic route beside Embden-Meyerhof-Parnas (EMP) and oxidative pentose phosphate (OPP) pathway in oxygenic photoautotrophs requires a revision of their central carbohydrate metabolism. In this study, unexpectedly, we observed that deletion of the ED pathway alone, and even more pronounced in combination with other glycolytic routes, diminished photoautotrophic growth in continuous light in the cyanobacterium Synechocystis sp. PCC 6803. Furthermore, we found that the ED pathway is required for optimal glycogen catabolism in parallel to an operating Calvin–Benson–Bassham (CBB) cycle. It is counter-intuitive that glycolytic routes, which are a reverse to the CBB cycle and do not provide any additional biosynthetic intermediates, are important under photoautotrophic conditions. However, observations on the ability to reactivate an arrested CBB cycle revealed that they form glycolytic shunts that tap the cellular carbohydrate reservoir to replenish the cycle. Taken together, our results suggest that the classical view of the CBB cycle as an autocatalytic, completely autonomous cycle that exclusively relies on its own enzymes and CO2 fixation to regenerate ribulose-1,5-bisphosphate for Rubisco is an oversimplification. We propose that in common with other known autocatalytic cycles, the CBB cycle likewise relies on anaplerotic reactions to compensate for the depletion of intermediates, particularly in transition states and under fluctuating light conditions that are common in nature