Given a bipartite density matrix ρ of a quantum state, the Quantum Separability problem (QUSEP) asks — is ρ entangled, or separable? In this thesis, we first strengthen Gurvits’ 2003 NP-hardness result for QUSEP by showing that the Weak Membership problem over the set of separable bipartite quantum states is strongly NP-hard, meaning it is NP-hard even when the error margin is as large as inverse polynomial in the dimension, i.e. is “moderately large”. Previously, this NP-hardness was known only to hold in the case of inverse exponential error. We observe the immediate implication of NP-hardness of the Weak Membership problem over the set of entanglement-breaking maps, as well as lower bounds on the maximum (Euclidean) distance possible between a bound entangled state and the separable set of quantum states (assuming P ≠ NP). We next investigate the entanglement-detecting capabilities of locally invariant unitary operations, as proposed by Fu in 2006. Denoting the subsystems of ρ as A and B, such that ρ_B = Tr_A(ρ), a locally invariant unitary operation U^B is one with the property U^B ρ_B (U^B)^† = ρ_B. We investigate the maximum shift (in Euclidean distance) inducible in ρ by applying I⊗U^B, over all locally invariant choices of U^B. We derive closed formulae for this quantity for three cases of interest: (pseudo)pure quantum states of arbitrary dimension, Werner states of arbitrary dimension, and two-qubit states. Surprisingly, similar to recent anomalies detected for non-locality measures, the first of these formulae demonstrates the existence of non-maximally entangled states attaining shifts as large as maximally entangled ones. Using the latter of these formulae, we demonstrate for certain classes of two-qubit states an equivalence between the Fu criterion and the CHSH inequality. Among other results, we investigate the ability of locally invariant unitary operations to detect bound entanglement.

Given a bipartite density matrix ρ of a quantum state, the Quantum Separability problem (QUSEP) asks — is ρ entangled, or separable? In this thesis, we first strengthen Gurvits’ 2003 NP-hardness result for QUSEP by showing that the Weak Membership problem over the set of separable bipartite quantum states is strongly NP-hard, meaning it is NP-hard even when the error margin is as large as inverse polynomial in the dimension, i.e. is “moderately large”. Previously, this NP-hardness was known only to hold in the case of inverse exponential error. We observe the immediate implication of NP-hardness of the Weak Membership problem over the set of entanglement-breaking maps, as well as lower bounds on the maximum (Euclidean) distance possible between a bound entangled state and the separable set of quantum states (assuming P ≠ NP). We next investigate the entanglement-detecting capabilities of locally invariant unitary operations, as proposed by Fu in 2006. Denoting the subsystems of ρ as A and B, such that ρ_B = Tr_A(ρ), a locally invariant unitary operation U^B is one with the property U^B ρ_B (U^B)^† = ρ_B. We investigate the maximum shift (in Euclidean distance) inducible in ρ by applying I⊗U^B, over all locally invariant choices of U^B. We derive closed formulae for this quantity for three cases of interest: (pseudo)pure quantum states of arbitrary dimension, Werner states of arbitrary dimension, and two-qubit states. Surprisingly, similar to recent anomalies detected for non-locality measures, the first of these formulae demonstrates the existence of non-maximally entangled states attaining shifts as large as maximally entangled ones. Using the latter of these formulae, we demonstrate for certain classes of two-qubit states an equivalence between the Fu criterion and the CHSH inequality. Among other results, we investigate the ability of locally invariant unitary operations to detect bound entanglement.

Orientador: Jose Roberto Nunhez / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Quimica / Made available in DSpace on 2018-08-06T09:02:08Z (GMT). No. of bitstreams: 1 Fernandes_LuisAntonioGalhego_M.pdf: 1106385 bytes, checksum: 6b2e2fa17d2a53c5051005b064878165 (MD5) Previous issue date: 2005 / Resumo: Com aplicações nos mais variados ramos industriais, tais como o alimentício, o químico, o farmacêutico, o de bebidas e o de toucador, entre outros, a operação unitária de agitação ou mistura de fluidos desempenha importante papel nos processos industriais. Os misturadores estáticos constituem uma alternativa aos vasos agitados tradicionais, sendo aplicados em processos contínuos. Os atuais fabricantes nacionais de misturadores estáticos têm apresentado dificuldade em definir o tipo adequado de misturador para as aplicações de um modo geral, como selecionar um misturador que, apesar de promover a mistura apresenta um alto consumo de energia para a aplicação. A constante preocupação em se reduzir o consumo de energia exige um cuidado especial com relação à escolha do misturador adequado ao processo. Dois misturadores denominados ALETAS e EDA foram selecionados de um trabalho anterior que desenvolveu e otimizou misturadores estáticos, utilizando a ferramenta fluido-dinâmica computacional (CFD), sendo que um destes se opera em regime laminar (ALETAS) e o outro em regime turbulento (EDA). O presente estudo buscou complementar o desenvolvimento destes novos tipos de misturadores estáticos, sob a forma de ensaios experimentais, em regimes laminar e turbulento. Como resultado dos ensaios experimentais pôde-se concluir que o misturador tipo ALETAS é adequado para se operar em regime laminar e o EDA pode ser utilizado em ambos regimes, laminar e turbulento, com alguns cuidados especiais para o caso de regime laminar. Os mesmos foram comparados com misturadores bastante estudados na literatura (Kenics e Sulzer SMX) apresentando desempenho similar a estes dois misturadores comerciais / Abstract: Mixing is a unit operation with applicability in many industrial fields, such as food, chemical, petrochemical, beverages and pharmaceutical, just to say some fields. The static mixers appear with an alternative to the traditional agitated vessels. They could be used such in batch mixing operation as in continuous processes, but mainly in continuous processes. The Brazilian suppliers had some problems in choosing the adequate equipment, and it is not rare that they choose an equipment that really does the proper mixture, but with a very high consumption of energy. The recently concern about saving energy demands a special care about the choosing of the best static mixer for the particular task.It was selected two models (ALETAS and EDA) from another work that developed static mixers using computational fluid-dynamic (CFD) method, the ALETAS mixer for laminar mixing and EDA for turbulent mixing. This work searches to complete this study, by experimental tests in laminar and turbulent mixing conditions.The results show that ALETAS static mixer can be used in laminar mixing, as the EDA static mixer but the EDA can be used in laminar mixing with some particular cares. The EDA can also be used in the turbulent mixing. Both models of static mixers were also compared with the Kenics and Sulzer SMX commercial static mixers using the data available in the literature with very similar results / Mestrado / Desenvolvimento de Processos Químicos / Mestre em Engenharia Química