Submitted by Renata Lopes (renatasil82@gmail.com) on 2018-09-03T15:49:55Z No. of bitstreams: 1 robertomalheirosmoreirafilho.pdf: 1784153 bytes, checksum: 65d1abd13eb713e62923e5d3c4d69acf (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2018-09-03T15:53:44Z (GMT) No. of bitstreams: 1 robertomalheirosmoreirafilho.pdf: 1784153 bytes, checksum: 65d1abd13eb713e62923e5d3c4d69acf (MD5) / Made available in DSpace on 2018-09-03T15:53:44Z (GMT). No. of bitstreams: 1 robertomalheirosmoreirafilho.pdf: 1784153 bytes, checksum: 65d1abd13eb713e62923e5d3c4d69acf (MD5) Previous issue date: 2012-07-30 / PROQUALI (UFJF) / A criação de regras de apoio à decisão com base em sistemas de informação é o objeto central da Teoria dos Conjuntos Aproximativos - TCA (Rough Sets Theory). Um trabalho apresentado por Pawlak em 1982 deu início a diversos estudos com o objetivo de criar regras de decisão baseadas em sistemas de informações com múltiplos atributos condicionantes e um ou mais atributos de decisão. Ao longo do tempo, os atributos com relação de dominância, onde há uma escala de valores, vêm recebendo destaque. Para lidar com este tipo de dados foi desenvolvido a DRSA (Dominance-based Rough Sets Approach). O excessivo rigor exigido para a criação de regras pela proposta básica do DRSA fez com que novas propostas surgissem. Além das regras determinísticas, com 100% de certeza, surgiram regras probabilísticas, com algum percentual controlado de incerteza. Existem algumas vertentes de estudo de aplicações de DRSA e a abordagem aqui proposta explora a possibilidade de aumento na qualidade da aproximação e, consequentemente, na qualidade das regras geradas, considerando a possibilidade de união de algumas classes do atributo de decisão com princípio de dominância. Com isto, são preservados os princípios do uso de DRSA. De acordo com a necessidade do pesquisador, a redução de classes pode ser utilizada em conjunto com as outras alternativas de DRSA apresentadas até o momento. Duas novas propostas de união de classes do atributo do critério de decisão são apresentadas, comentadas e criticadas nesta tese, uma baseada em função densidade de probabilidades e outra baseada em transformações probabilísticas. / Creating rules for the support of the decision process is the main subject of Rough Sets Theory. The study first published by Pawlak in 1982 was a catalyst of several studies focusing on creating rules for the support of the decision making process based on multiple attributes conditioning one or more decision attributes. As the studies evolved, attributes with the feature of dominance – attributes measured in some scale – have caught attention. This gave rise to DRSA (Dominance-based Rough Sets Approach). The excessively strict guidelines of DRSA original studies led to the creation of new guidelines that consider not only deterministic rules – 100% certainty – but also probabilistic rules, which account for a certain degree of uncertainty. There are other developments of DRSA and the approach here taken evaluates alternatives for enhancing the quality of the approximation evaluation, therefore enhancing the quality of the rules, by clustering classes of values of decision attributes without compromising the guiding principles of DRSA. According to the need of researcher, the reduction of classes can be used in conjunction with other alternatives of development of DRSA. Two different proposals for the clustering of attributes are presented and evaluated in this study, one based on density functions and the other based on probabilistic transformations.

The processing of information based on the generation of common quantum optical states (e.g. coherent states) and the measurement of the quadrature components of the light field (e.g. homodyne detection) is often referred to as continuous-variable quantum information processing. It is a very fertile field of investigation, at a crossroads between quantum optics and information theory, with notable successes such as unconditional continuous-variable quantum teleportation or Gaussian quantum key distribution. In quantum optics, the states of the light field are conveniently characterized using a phase-space representation (e.g. Wigner function), and the common optical components effect simple affine transformations in phase space (e.g. rotations). In quantum information theory, one often needs to determine entropic characteristics of quantum states and operations, since the von Neuman entropy is the quantity at the heart of entanglement measures or channel capacities. Computing entropies of quantum optical states requires instead turning to a state-space representation of the light field, which formally is the Fock space of a bosonic mode.<p>This interplay between phase-space and state-space representations does not represent a particular problem as long as Gaussian states (e.g. coherent, squeezed, or thermal states) and Gaussian operations (e.g. beam splitters or squeezers) are concerned. Indeed, Gaussian states are fully characterized by the first- and second-order moments of mode operators, while Gaussian operations are defined via their actions on these moments. The so-called symplectic formalism can be used to treat all Gaussian transformations on Gaussian states, including mixed states of an arbitrary number of modes, and the entropies of Gaussian states are directly linked to their symplectic eigenvalues.<p>This thesis is concerned with the Gaussian transformations applied onto arbitrary states of light, in which case the symplectic formalism is unapplicable and this phase-to-state space interplay becomes highly non trivial. A first motivation to consider arbitrary (non-Gaussian) states of light results from various Gaussian no-go theorems in continuous-variable quantum information theory. For instance, universal quantum computing, quantum entanglement concentration, or quantum error correction are known to be impossible when restricted to the Gaussian realm. A second motivation comes from the fact that several fundamental quantities, such as the entanglement of formation of a Gaussian state or the communication capacity of a Gaussian channel, rely on an optimization over all states, including non-Gaussian states even though the considered state or channel is Gaussian. This thesis is therefore devoted to developing new tools in order to compute state-space properties (e.g. entropies) of transformations defined in phase-space or conversely to computing phase-space properties (e.g. mean-field amplitudes) of transformations defined in state space. Remarkably, even some basic questions such as the entanglement generation of optical squeezers or beam splitters were unsolved, which gave us a nice work-bench to investigate this interplay. <p>In the first part of this thesis (Chapter 3), we considered a recently discovered Gaussian probabilistic transformation called the noiseless optical amplifier. More specifically, this is a process enabling the amplification of a quantum state without introducing noise. As it has long been known, when amplifing a quantum signal, the arising of noise is inevitable due to the unitary evolution that governs quantum mechanics. It was recently realized, however, that one can drop the unitarity of the amplification procedure and trade it for a noiseless, albeit probabilistic (heralded) transformation. The fact that the transformation is probabilistic is mathematically reflected in the fact that it is non trace-preserving. This quantum device has gained much interest during the last years because it can be used to compensate losses in a quantum channel, for entanglement distillation, probabilistic quantum cloning, or quantum error correction. Several experimental demonstrations of this device have already been carried out. Our contribution to this topic has been to derive the action of this device on squeezed states and to prove that it acts quite surprisingly as a universal (phase-insensitive) optical squeezer, conserving the signal-to-noise ratio just as a phase-sensitive optical amplifier but for all quadratures at the same time. This also brought into surface a paradoxical effect, namely that such a device could seemingly lead to instantaneous signaling by circumventing the quantum no-cloning theorem. This paradox was discussed and resolved in our work.<p>In a second step, the action of the noiseless optical amplifier and it dual operation (i.e. heralded noiseless attenuator) on non-Gaussian states has been examined. We have observed that the mean-field amplitude may decrease in the process of noiseless amplification (or may increase in the process of noiseless attenuation), a very counterintuitive effect that Gaussian states cannot exhibit. This work illustrates the above-mentioned phase-to-state space interplay since these devices are defined as simple filtering operations in state space but inferring their action on phase-space quantities such as the mean-field amplitude is not straightforward. It also illustrates the difficulty of dealing with non-Gaussian states in Gaussian transformations (these noiseless devices are probabilistic but Gaussian). Furthermore, we have exhibited an experimental proposal that could be used to test this counterintuitive feature. The proposed set-up is feasible with current technology and robust against usual inefficiencies that occur in optical experiment. <p>Noiseless amplification and attenuation represent new important tools, which may offer interesting perspectives in quantum optical communications. Therefore, further understanding of these transformations is both of fundamental interest and important for the development and analysis of protocols exploiting these tools. Our work provides a better understanding of these transformations and reveals that the intuition based on ordinary (deterministic phase-insensitive) amplifiers and losses is not always applicable to the noiseless amplifiers and attenuators.<p>In the last part of this thesis, we have considered the entropic characterization of some of the most fundamental Gaussian transformations in quantum optics, namely a beam splitter and two-mode squeezer. A beam splitter effects a simple rotation in phase space, while a two-mode squeezer produces a Bogoliubov transformation. Thus, there is a well-known phase-space characterization in terms of symplectic transformations, but the difficulty originates from that one must return to state space in order to access quantum entropies or entanglement. This is again a hard problem, linked to the above-mentioned interplay in the reverse direction this time. As soon as non-Gaussian states are concerned, there is no way of calculating the entropy produced by such Gaussian transformations. We have investigated two novel tools in order to treat non-Gaussian states under Gaussian transformations, namely majorization theory and the replica method.<p>In Chapter 4, we have started by analyzing the entanglement generated by a beam splitter that is fed with a photon-number state, and have shown that the entanglement monotones can be neatly combined with majorization theory in this context. Majorization theory provides a preorder relation between bipartite pure quantum states, and gives a necessary and sufficient condition for the existence of a deterministic LOCC (local operations and classical communication) transformation from one state to another. We have shown that the state resulting from n photons impinging on a beam splitter majorizes the corresponding state with any larger photon number n’ > n, implying that the entanglement monotonically grows with n, as expected. In contrast, we have proven that such a seemingly simple optical component may have a rather surprising behavior when it comes to majorization theory: it does not necessarily lead to states that obey a majorization relation if one varies the transmittance (moving towards a balanced beam splitter). These results are significant for entanglement manipulation, giving rise in particular to a catalysis effect.<p>Moving forward, in Chapter 5, we took the step of introducing the replica method in quantum optics, with the goal of achieving an entropic characterization of general Gaussian operations on a bosonic quantum field. The replica method, a tool borrowed from statistical physics, can also be used to calculate the von Neumann entropy and is the last line of defense when the usual definition is not practical, which is often the case in quantum optics since the definition involves calculating the eigenvalues of some (infinite-dimensional) density matrix. With this method, the entropy produced by a two-mode squeezer (or parametric optical amplifier) with non-trivial input states has been studied. As an application, we have determined the entropy generated by amplifying a binary superposition of the vacuum and an arbitrary Fock state, which yields a surprisingly simple, yet unknown analytical expression. Finally, we have turned to the replica method in the context of field theory, and have examined the behavior of a bosonic field with finite temperature when the temperature decreases. To this end, information theoretical tools were used, such as the geometric entropy and the mutual information, and interesting connection between phase transitions and informational quantities were found. More specifically, dividing the field in two spatial regions and calculating the mutual information between these two regions, it turns out that the mutual information is non-differentiable exactly at the critical temperature for the formation of the Bose-Einstein condensate.<p>The replica method provides a new angle of attack to access quantum entropies in fundamental Gaussian bosonic transformations, that is quadratic interactions between bosonic mode operators such as Bogoliubov transformations. The difficulty of accessing entropies produced when transforming non-Gaussian states is also linked to several currently unproven entropic conjectures on Gaussian optimality in the context of bosonic channels. Notably, determining the capacity of a multiple-access or broadcast Gaussian bosonic channel is pending on being able to access entropies. We anticipate that the replica method may become an invaluable tool in order to reach a complete entropic characterization of Gaussian bosonic transformations, or perhaps even solve some of these pending conjectures on Gaussian bosonic channels.<p> / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished