Efficient Characterisation and Optimal Control of Open Quantum Systems - Mathematical Foundations and Physical Applications

Since no physical system can ever be completely isolated from its environment, the study of open quantum systems is pivotal to reliably and accurately control complex quantum systems. In practice, reliability of the control field needs to be confirmed via certification of the target evolution while accuracy requires the derivation of high-fidelity control schemes in the presence of decoherence. In the first part of this thesis an algebraic framework is presented that allows to determine the minimal requirements on the unique characterisation of arbitrary unitary gates in open quantum systems, independent on the particular physical implementation of the employed quantum device. To this end, a set of theorems is devised that can be used to assess whether a given set of input states on a quantum channel is sufficient to judge whether a desired unitary gate is realised. This allows to determine the minimal input for such a task, which proves to be, quite remarkably, independent of system size. These results allow to elucidate the fundamental limits regarding certification and tomography of open quantum systems. The combination of these insights with state-of-the-art Monte Carlo process certification techniques permits a significant improvement of the scaling when certifying arbitrary unitary gates. This improvement is not only restricted to quantum information devices where the basic information carrier is the qubit but it also extends to systems where the fundamental informational entities can be of arbitary dimensionality, the so-called qudits. The second part of this thesis concerns the impact of these findings from the point of view of Optimal Control Theory (OCT). OCT for quantum systems utilises concepts from engineering such as feedback and optimisation to engineer constructive and destructive interferences in order to steer a physical process in a desired direction. It turns out that the aforementioned mathematical findings allow to deduce novel optimisation functionals that significantly reduce not only the required memory for numerical control algorithms but also the total CPU time required to obtain a certain fidelity for the optimised process. The thesis concludes by discussing two problems of fundamental interest in quantum information processing from the point of view of optimal control - the preparation of pure states and the implementation of unitary gates in open quantum systems. For both cases specific physical examples are considered: for the former the vibrational cooling of molecules via optical pumping and for the latter a superconducting phase qudit implementation. In particular, it is illustrated how features of the environment can be exploited to reach the desired targets.

The physical properties of the atmosphere in the new Hadley Centre Global Environmental Model (HadGEM1). Part II: Aspects of variability and regional climate

The performance of the atmospheric component of the new Hadley Centre Global Environmental Model (HadGEM1) is assessed in terms of its ability to represent a selection of key aspects of variability in the Tropics and extratropics. These include midlatitude storm tracks and blocking activity, synoptic variability over Europe, and the North Atlantic Oscillation together with tropical convection, the Madden-Julian oscillation, and the Asian summer monsoon. Comparisons with the previous model, the Third Hadley Centre Coupled Ocean-Atmosphere GCM (HadCM3), demonstrate that there has been a considerable increase in the transient eddy kinetic energy (EKE), bringing HadGEM1 into closer agreement with current reanalyses. This increase in EKE results from the increased horizontal resolution and, in combination with the improved physical parameterizations, leads to improvements in the representation of Northern Hemisphere storm tracks and blocking. The simulation of synoptic weather regimes over Europe is also greatly improved compared to HadCM3, again due to both increased resolution and other model developments. The variability of convection in the equatorial region is generally stronger and closer to observations than in HadCM3. There is, however, still limited convective variance coincident with several of the observed equatorial wave modes. Simulation of the Madden-Julian oscillation is improved in HadGEM1: both the activity and interannual variability are increased and the eastward propagation, although slower than observed, is much better simulated. While some aspects of the climatology of the Asian summer monsoon are improved in HadGEM1, the upper-level winds are too weak and the simulation of precipitation deteriorates. The dominant modes of monsoon interannual variability are similar in the two models, although in HadCM3 this is linked to SST forcing, while in HadGEM1 internal variability dominates. Overall, analysis of the phenomena considered here indicates that HadGEM1 performs well and, in many important respects, improves upon HadCM3. Together with the improved representation of the mean climate, this improvement in the simulation of atmospheric variability suggests that HadGEM1 provides a sound basis for future studies of climate and climate change.

Response theory for equilibrium and non-equilibrium statistical mechanics: causality and generalized Kramers-Kronig relations

We consider the general response theory recently proposed by Ruelle for describing the impact of small perturbations to the non-equilibrium steady states resulting from Axiom A dynamical systems. We show that the causality of the response functions entails the possibility of writing a set of Kramers-Kronig (K-K) relations for the corresponding susceptibilities at all orders of nonlinearity. Nonetheless, only a special class of directly observable susceptibilities obey K-K relations. Specific results are provided for the case of arbitrary order harmonic response, which allows for a very comprehensive K-K analysis and the establishment of sum rules connecting the asymptotic behavior of the harmonic generation susceptibility to the short-time response of the perturbed system. These results set in a more general theoretical framework previous findings obtained for optical systems and simple mechanical models, and shed light on the very general impact of considering the principle of causality for testing self-consistency: the described dispersion relations constitute unavoidable benchmarks that any experimental and model generated dataset must obey. The theory exposed in the present paper is dual to the time-dependent theory of perturbations to equilibrium states and to non-equilibrium steady states, and has in principle similar range of applicability and limitations. In order to connect the equilibrium and the non equilibrium steady state case, we show how to rewrite the classical response theory by Kubo so that response functions formally identical to those proposed by Ruelle, apart from the measure involved in the phase space integration, are obtained. These results, taking into account the chaotic hypothesis by Gallavotti and Cohen, might be relevant in several fields, including climate research. In particular, whereas the fluctuation-dissipation theorem does not work for non-equilibrium systems, because of the non-equivalence between internal and external fluctuations, K-K relations might be robust tools for the definition of a self-consistent theory of climate change.

On Aspects of Mathematical Reasoning : Affect and Gender

This thesis explores two aspects of mathematical reasoning: affect and gender. I started by looking at the reasoning of upper secondary students when solving tasks. This work revealed that when not guided by an interviewer, algorithmic reasoning, based on memorising algorithms which may or may not be appropriate for the task, was predominant in the students reasoning. Given this lack of mathematical grounding in students reasoning I looked in a second study at what grounds they had for different strategy choices and conclusions. This qualitative study suggested that beliefs about safety, expectation and motivation were important in the central decisions made during task solving.  But are reasoning and beliefs gendered? The third study explored upper secondary school teachers conceptions about gender and students mathematical reasoning. In this study I found that upper secondary school teachers attributed gender symbols including insecurity, use of standard methods and imitative reasoning to girls and symbols such as multiple strategies especially on the calculator, guessing and chance-taking were assigned to boys. In the fourth and final study I found that students, both male and female, shared their teachers view of rather traditional feminities and masculinities. Remarkably however, this result did not repeat itself when students were asked to reflect on their own behaviour: there were some discrepancies between the traits the students ascribed as gender different and the traits they ascribed to themselves. Taken together the thesis suggests that, contrary to conceptions, girls and boys share many of the same core beliefs about mathematics, but much work is still needed if we should create learning environments that provide better opportunities for students to develop beliefs that guide them towards well-grounded mathematical reasoning. 

Obesity: effects of physical exercise and metformin on metabolic aspects of monosodium glutamate treated rats

The present study investigated the effects of swimming training and metformin on metabolic aspects of obese rats. Wistar rats were divided into control (C), obese (O), Trained Obese (TO) and metformin obese (MO) groups. Obesity was induced by subcutaneous monosodium glutamate injection (4 mg/g body weight). Exercise program consisted in swimming 1 h/day, 5 days/week, for 8 weeks, supporting a load corresponding to 5% of body weight. Metformin was dissolved in the drinking water (1.4 mg/ml) for 8 weeks. At the end of the experimental period, rats were sacrificed and blood was collected for determinations of serum glucose, insulin and triglycerides and hematocrit. Samples of gastrocnemius muscle and liver were removed to evaluate triglycerides content MSG-induced obesity, increased serum glucose, insulin and triglycerides, while physical training was able to recover serum glucose and insulin and metformin treatment recovered serum insulin and slightly reduced the serum glucose. MSG-induced obesity also increased liver triglycerides content and physical training and metformin administration recovered these parameters. It was concluded that in MSG obese rats, physical exercise and metformin induced important metabolic alterations associated with an improvement in glucose homeostasis and in liver fat content. Obesity and Metabolism 2009; 5: 129-133.

Relation between the compaction rate and physical and mechanical properties of particleboards

The compaction rate, the relation between the density of the wood panel and the density of the wood used for producing the particles, is an indicator of the product's densification. Among the various types of wood panels, particleboards are widely employed in the lumber industry, mainly for the furniture production. This paper presents a study of the relation between the compaction rate and the properties of tensile strength perpendicular to surface, Modulus of Rupture (MOR) and Modulus of Elasticity (MOE) obtained from a static bending test, thickness swelling and water absorption (2 and 24 hours). These properties were calculated according to the Brazilian ABNT, NBR 14810 standard. Particleboards were produced using the species Pinus elliotti and adhesive ureaformaldehyde. The relation was established by a multiple linear regression, and the most appropriate statistical models were determined. The estimated models indicate statistically significant effects of water absorption in 2 hours and MOR in the particleboards' compaction rate.

Monitoring chronic physical stress using biomarkers, performance protocols and mathematical functions to identify physiological adaptations in rats

This study was undertaken to characterize the effects of monotonous training at lactate minimum (LM) intensity on aerobic and anaerobic performances; glycogen concentrationsin the soleus muscle, the gastrocnemius muscle and the liver; and creatine kinase (CK), free fatty acids and glucose concentrations in rats. The rats were separated into trained (n =10), baseline (n = 10) and sedentary (n=10) groups. The trained group was submitted to the following: 60 min/day, 6 day/week and intensity equivalent to LM during the 12-week training period. The training volume was reduced after four weeks according to a sigmoid function. The total CK (U/L) increased in the trained group after 12 weeks (742.0±158.5) in comparison with the baseline (319.6±40.2) and the sedentary (261.6+42.2) groups. Free fatty acids and glycogen stores (liver, soleus muscle and gastrocnemius muscle) increased after 12 weeks of monotonous training but aerobic and anaerobic performances were unchanged in relation to the sedentary group. The monotonous training at LM increased the level of energy substrates, unchanged aerobic performance, reduced anaerobic capacity and increased the serum CK concentration; however, the rats did not achieve the predicted training volume.

Semi-synthetic minimal cells: Biochemical, physical, and technological aspects

The construction of synthetic cells is one of the major goals of bioengineering. The most successful approach consists in the encapsulation of biochemical materials (DNA, RNA, enzymes, etc.) inside lipid vesicles (liposomes), mimicking a cell structure. In this contribution, that also aims at introducing the reader to 'chemical synthetic biology,' we describe the current state of the art of 'semi-synthetic minimal cells' (SSMCs), namely, cell-like structures containing the minimal number of biological compounds that are required to reconstruct a function of interest. We will first describe how the concept of the minimal cell was originated and its relation with the theory of autopoiesis, then we review the most advanced results focused on genetic/metabolic networks inside liposomes. Next, we emphasize that relevance of physical aspects (too often neglected) that impact on the solute entrapment process, and finally we discuss new technological trends in SSMC research that will probably allow their future use in biotechnology. © 2013 Copyright © 2013 Elsevier Inc. All rights reserved.

Some mathematical aspects and scattering amplitudes in the pure spinor formalism

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Statistical Mechanics of Monomer-Dimer Models on Complete and Erdös-Rényi Graphs

Nella tesi sono trattate due famiglie di modelli meccanico statistici su vari grafi: i modelli di spin ferromagnetici (o di Ising) e i modelli di monomero-dimero. Il primo capitolo è dedicato principalmente allo studio del lavoro di Dembo e Montanari, in cui viene risolto il modello di Ising su grafi aleatori. Nel secondo capitolo vengono studiati i modelli di monomero-dimero, a partire dal lavoro di Heilemann e Lieb,con l'intento di dare contributi nuovi alla teoria. I principali temi trattati sono disuguaglianze di correlazione, soluzioni esatte su alcuni grafi ad albero e sul grafo completo, la concentrazione dell'energia libera intorno al proprio valor medio sul grafo aleatorio diluito di Erdös-Rényi.

Statistical mechanics formalism and methods for the analysis of real networks

This thesis provides a thoroughly theoretical background in network theory and shows novel applications to real problems and data. In the first chapter a general introduction to network ensembles is given, and the relations with “standard” equilibrium statistical mechanics are described. Moreover, an entropy measure is considered to analyze statistical properties of the integrated PPI-signalling-mRNA expression networks in different cases. In the second chapter multilayer networks are introduced to evaluate and quantify the correlations between real interdependent networks. Multiplex networks describing citation-collaboration interactions and patterns in colorectal cancer are presented. The last chapter is completely dedicated to control theory and its relation with network theory. We characterise how the structural controllability of a network is affected by the fraction of low in-degree and low out-degree nodes. Finally, we present a novel approach to the controllability of multiplex networks

Statistical mechanics of polymer models: rigorous results and applications

Monomer-dimer models are amongst the models in statistical mechanics which found application in many areas of science, ranging from biology to social sciences. This model describes a many-body system in which monoatomic and diatomic particles subject to hard-core interactions get deposited on a graph. In our work we provide an extension of this model to higher-order particles. The aim of our work is threefold: first we study the thermodynamic properties of the newly introduced model. We solve analytically some regular cases and find that, differently from the original, our extension admits phase transitions. Then we tackle the inverse problem, both from an analytical and numerical perspective. Finally we propose an application to aggregation phenomena in virtual messaging services.

New aspects of airway mechanics in pre-term infants

High-frequency respiratory impedance data measured noninvasively by the high-speed interrupter technique (HIT), particularly the first antiresonance frequency (f(ar,1)), is related to airway wall mechanics. The aim of this study was to evaluate the feasibility and repeatability of HIT in unsedated pre-term infants, and to compare values of f(ar,1) from 18 pre-term (post-conceptional age 32-37 weeks, weight 1,730-2,910 g) and 18 full-term infants (42-47 weeks, 3,920-5,340 g). Among the pre-term infants, there was good short-term repeatability of f(ar,1) within a single sleep epoch (mean (sd) coefficient of variance: 8 (1.7)%), but 95% limits of agreement for repeated measures of f(ar,1) after 3-8 h were relatively wide (-41 Hz; 37 Hz). f(ar,1) was significantly lower in pre-term infants (199 versus 257 Hz), indicating that wave propagation characteristics in pre-term airways are different from those of full-term infants. The present authors suggest that this is consistent with developmental differences in airway wall structure and compliance, including the influence of the surrounding tissue. Since flow limitation is determined by wave propagation velocity and airway cross-sectional area, it was hypothesised that the physical ability of the airways to carry large flows is fundamentally different in pre-term than in full-term infants.