Depuis l’introduction de la mécanique quantique, plusieurs mystères de la nature ont trouvé leurs explications. De plus en plus, les concepts de la mécanique quantique se sont entremêlés avec d’autres de la théorie de la complexité du calcul. De nouvelles idées et solutions ont été découvertes et élaborées dans le but de résoudre ces problèmes informatiques. En particulier, la mécanique quantique a secoué plusieurs preuves de sécurité de protocoles classiques. Dans ce m´emoire, nous faisons un étalage de résultats récents de l’implication de la mécanique quantique sur la complexité du calcul, et cela plus précisément dans le cas de classes avec interaction. Nous présentons ces travaux de recherches avec la nomenclature des jeux à information imparfaite avec coopération. Nous exposons les différences entre les théories classiques, quantiques et non-signalantes et les démontrons par l’exemple du jeu à cycle impair. Nous centralisons notre attention autour de deux grands thèmes : l’effet sur un jeu de l’ajout de joueurs et de la répétition parallèle. Nous observons que l’effet de ces modifications a des conséquences très différentes en fonction de la théorie physique considérée. / Since the introduction of quantum mechanics, many mysteries of nature have found explanations. Many quantum-mechanical concepts have merged with the field of computational complexity theory. New ideas and solutions have been put forward to solve computational problems. In particular, quantum mechanics has struck down many security proofs of classical protocols. In this thesis, we survey recent results regarding the implication of quantum mechanics to computational complexity and more precisely to classes with interaction. We present the work done in the framework of cooperative games with imperfect information. We give some differences between classical, quantum and no-signaling theories and apply them to the specific example of Odd Cycle Games. We center our attention on two different themes: the effect on a game of adding more players and of parallel repetition. We observe that depending of the physical theory considered, the consequences of these changes is very different.

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[pt] Uma propriedade notável dos sistemas biológicos é a formação de estruturas espaciais. Estas podem surgir por auto-organização, como consequência das próprias interações entre os indivíduos. Para estudar estas estruturas e como elas emergem, têm sido muito úteis modelos simples para a dinâmica da densidade espacial de uma população, que levam em conta apenas certos processos elementares (como reprodução, competição e dispersão). Em particular, a equação de FKPP (Fisher-Kolmogorov- Petrovski-Piskunov), que inclui simplesmente o crescimento logístico mais a difusão normal, é um modelo clássico para a dinâmica de uma população de uma única espécie. Dentro do quadro minimalista da equação de FKPP e suas variantes, a competição à distância (ou, não local) é a principal responsável por produzir oscilações espaciais na densidade da população. Entretanto, a não localidade pode ocorrer também nos demais processos. Assim, um primeiro objetivo desta tese é investigar como as diferentes escalas espaciais presentes podem interferir entre si, afetando a formação de padrões. Para isso, consideramos uma generalização da equação de FKPP em que todos os termos são não locais, em um ambiente homogêneo com condições de contorno periódicas. Enquanto a competição é o principal processo por trás da formação de padrões, mostramos que os outros dois podem agir de forma construtiva ou destrutiva. Por exemplo, a difusão, que comumente homogeniza, pode favorecer a formação de padrões dependendo do formato e alcance das funções de influência de cada processo. Em um segundo estudo, motivado por resultados experimentais, procuramos entender como a variabilidade da difusividade pode impactar a organização espacial da população dentro e fora de um refúgio (região de alta qualidade imersa em um ambiente hostil). Para tanto, consideramos uma outra generalização da equação de FKPP, com não localidade apenas no processo de competição intra-espécie, e modificada para levar em conta a presença do refúgio. Além da dependência espacial da taxa de crescimento, que é a principal característica distintiva de um refúgio em um ambiente hostil, também consideramos o fato de que a mobilidade pode ser heterogênea no espaço ou depender da densidade populacional. Focamos em dois casos em que a difusividade responde à densidade de indivíduos, diminuindo ou aumentando com a densidade populacional. Para comparação, também abordamos a difusividade dependente do espaço, com valores diferentes dentro e fora do refúgio. Observamos que o limiar da formação de padrões, no espaço de parâmetros, é bastante robusto diante destas heterogeneidades. Por outro lado, a dependência com a densidade pode produzir uma realimentação que está ausente em meios homogêneos, e que afeta a forma dos padrões. Em todos os casos, os resultados foram obtidos mediante a integração numérica das equações integro-diferenciais e realizando considerações analíticas. / [en] A remarkable property of biological systems is the formation of spatial structures. These can arise by self-organization, as a consequence of the interactions between individuals. To study these structures and how they emerge, simple models for the dynamics of the spatial density of a population, which take into account only certain elementary processes (such as reproduction, competition and dispersion) have been very useful. In particular, the FKPP (Fisher-Kolmogorov-Petrovski-Piskunov) equation, which simply includes logistic growth plus normal diffusion, is a classic model for the dynamics of a population of a single species. Within the minimalist framework of the FKPP equation and its variants, distance (or, non-local) competition is primarily responsible for producing spatial oscillations in population density. However, non-locality can also be present in other processes. Then, a first objective of this thesis is to investigate how the different spatial scales which are present in each process can interfere between them, affecting the formation of patterns in a homogeneous environment with periodic boundary conditions. For this purpose, we consider a generalization of the FKPP equation in which all terms are nonlocal. While competition is the main process behind pattern formation, we show that the other two can act constructively or destructively. For example, diffusion, which commonly homogenizes, can favor the formation of patterns depending on the format and range of the influence functions of each process. In a second study, motivated by experimental results, we seek to understand how the variability of the diffusivity can impact the spatial organization of the population inside and outside a refuge (a high-quality region immersed in a hostile environment). Therefore, we consider another generalization of the FKPP equation, with non-locality only in the intra-species competition process, modified to take into account the presence of the refuge. In addition to the spatial dependence of the growth rate, which is the main distinguishing feature of a refuge in a hostile environment, we also consider the fact that mobility can be spatially heterogeneous or depend on population density. We focus on two cases in which diffusivity responds to the density of individuals, decreasing or increasing with population density. For comparison, we also address spacedependent diffusivity, with different values inside and outside the refuge. We observed that the threshold of pattern formation in parameter space is quite robust under the presence of these heterogeneities. On the other hand, density dependence can produce a feedback that is absent in homogeneous media, and that affects the shape of the patterns. In all cases, the results were obtained by numerical simulations of the integro-differential equations and by analytical considerations.

In dieser Arbeit werden die nichtlokalen sowie nichtlinearen Eigenschaften plasmonischer Nanopartikel behandelt, wie sie im hydrodynamischen Modell enthalten sind. Das hydrodynamische Materialmodell stellt eine Erweiterung des Drude Modells dar, in der Korrekturen in der Beschreibung des Elektronenplasmas berücksichtigt werden. Einer ausführlichen Einführung des Materialmodells folgt eine analytische Diskussion der Auswirkungen der Nichtlokalität am Beispiel eines einzelnen Zylinders. Hierbei werden die durch die Nichtlokalität herbeigeführten Frequenzverschiebungen in den Streu- und Absorptionsspektren quantifiziert und asymptotisch behandelt. Des Weiteren wird mit Hilfe einer konformen Abbildung das Problem eines zylindrischen Dimers in der Elektrostatischen Näherung gelöst und die Moden der Struktur bestimmt. Diese Untersuchungen dienen als maßgebliche Grundlage für weiterführende numerische Studien die mit der diskontinuierlichen Galerkin Zeitraummethode durchgeführt werden. Die durch die analytischen Betrachtungen gewonnene Kenntnis der Moden ermöglicht es, im Zusammenhang mit gruppentheoretischen Betrachtungen und numerischen Untersuchungen, rigorose Auswahlregeln für die Anregung der Moden durch lineare und nichtlineare Prozesse aufzustellen. In weiterführenden numerischen Simulationen werden außerdem Strukturen niedrigerer Symmetrie, auf die sich die Auswahlregeln übertragen lassen, untersucht. Zudem werden numerische Studien präsentiert in denen der Einfluss der Nichtlokalität auf Feldüberhöhungen in Dimeren und doppel-resonantes Verhalten (es liegt sowohl bei der Frequenz des eingestrahlten Lichtes als auch bei der zweiten harmonischen eine Resonanz vor) untersucht werden. / This thesis deals with the nonlocal and nonlinear properties of plasmonic nanoparticles, as described by the hydrodynamic model. The hydrodynamic material model represents an extension of the Drude model that contains corrections to the descriptions of the electron plasma. After a thorough derivation of the material model, analytical discussions of nonlocality are presented for the example of a single cylinder. The frequency shifts in the scattering and absorption spectra are quantified and treated asymptotically. Furthermore, by applying a conformal map, the problem of a cylindrical dimer is solved in the electrostatic limit and the modes of the structure are determined. These investigations lay the foundations for numerical investigations which are performed employing the discontinuous Galerkin time domain method. The analytical knowledge of the modes, in conjunction with group theoretical considerations and numerical analysis, enables the formulation of rigorous selection rules for the excitation of modes by linear and nonlinear processes. In further numerical studies, the influence of nonlocality on the field enhancement in dimer structures and double-resonant behavior (a resonance is found at the frequency of the incoming light and at the second harmonic) are investigated.

Quantum cryptography is arguably the most successfully applied area of quantum information theory. In this work, We invsetigate the role of quantum indistinguishability in random number generation, quantum temporal correlations, quantum nonlocality and counterfactuality for quantum cryptography. We study quantum protocols for key distribution, and their security in the conventional setting, in the counterfactual paradigm, and finally also in the device-independent scenario as applied to prepare-and-measure schemes. We begin with the interplay of two essential non-classical features like quantum indeterminism and quantum indistinguishability via a process known as bosonic stimulation is discussed. It is observed that the process provides an efficient method for macroscopic extraction of quantum randomness. Next, we propose two counterfactual cryptographic protocols, in which a secret key bit is generated even without the physical transmission of a particle. The first protocol is semicounterfactual in the sense that only one of the key bits is generated using interaction-free measurement. This protocol departs fundamentally from the original counterfactual key distribution protocol in not encoding secret bits in terms of photon polarization. We discuss how the security in the protocol originates from quantum single-particle non-locality. The second protocol is designed for the crypto-task of certificate authorization, where a trusted third party authenticates an entity (e.g., bank) to a client. We analyze the security of both protocols under various general incoherent attack models. The next part of our work includes study of quantum temporal correlations. We consider the use of the Leggett-Garg inequalities for device-independent security appropriate for prepare-and-measure protocols subjected to the higher dimensional attack that would completely undermine standard BB84. In the last part, we introduce the novel concept of nonlocal subspaces constructed using the graph state formalism, and propose their application for quantum information splitting. In particular, we use the stabilizer formalism of graph states to construct degenerate Bell operators, whose eigenspace determines the nonlocal subspace, into which a quantum secret is encoded and shared among an authorized group of agents, or securely transmitted to a designated secret retriever. The security of our scheme arises from the monogamy of quantum correlations. The quantum violation of the Bell-type inequality here is to its algebraic maximum, making this approach inherently suitable for the device-independent scenario.