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Geometrically Induced Singular Behavior of Entanglement Daniel Cavalcanti 1, Fernando G. S. L. Brandão 2, Pablo L. Saldanha 3, Olavo Cosme 3, Carlos H. Monken 3, Sebastião Pádua 3, Marcelo F. Santos 3, Marcelo O. Terra Cunha 4 1 ICFO-Institut de Ciencies Fotoniques, Spain 2 Imperial College London, England 3 Departamento de Física, Universidade Federal de Minas Gerais, Brasil 4 Departamento de Matemática, Universidade Federal de Minas Gerais, Brasil

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R R ( ) = Minimum s such that the state below is separable: R R can be used to investigate the shape of the set of separable states!!!! Random Robustness [1]: Quantifying Entanglement: [1] G.Vidal and R. Tarrack, Phys. Rev. A 59,141 (1999)

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Proposition: If the border of the set of separable states is a C m manifold (i.e. it can be parametrized by functions with continuous m th derivatives) and the state (q) describe an analytic curve in the space of states, then R R ( (q)) is also a C m function. Singularities in the border of the set of separable states will show up as singularities in R R ( (q)).

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A Hermitian operator W with positive mean value for all separable states. The optimal entanglement witness W opt to the state is the one that maximises |Tr{W }|. Tr{W }<0 is entangled Entanglement Witness [2]: For the case of two qubits, we have [3]: [3] F. G. S. L. Brandão and R. O. Vianna, Int. J. Quant. Inf 4, 331 (2006) The geometry of the set of quantum states is related to the mean value of a observable!!!!!! [2] M. Horodecki, P. Horodecki and R. Horodecki, Phys. Lett. A 223, 1 (1996)

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Constructing the state whereand We have

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Where does this state appears? Supose we have a system of four qubits a, b, A e B subject to the Hamiltonian: where and If the initial state is and we look to the temporal evolution for the subsystem AB, tracing out the subsystem ab, we have: [4] with [4] D.Cavalcanti et. al., Phys. Rev. A 74, (2006)

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Experimental Setup: [5] P. G. Kwiat et. al., Phys. Rev. Lett. 75, 4337 (1995) [5]

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Measuring R R ( )=-Tr{W opt )

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Conclusions: We have presented a method for probing the shape of the separable statesset. This method was implemented and singularities in this set were found. It must be emphasized that the present ideas can be generalized to other kinds of entanglement and to higher dimensional systems.

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