1. Geometrically Induced Singular Behavior of Entanglement
Daniel Cavalcanti1, Fernando G. S. L. Brandão2, Pablo L. Saldanha3, Olavo Cosme3 , Carlos H. Monken3, Sebastião Pádua3, Marcelo F. Santos3, Marcelo O. Terra Cunha4
1ICFO-Institut de Ciencies Fotoniques, Spain
2Imperial College London, England
3Departamento de Física, Universidade Federal de Minas Gerais, Brasil
4Departamento de Matemática, Universidade Federal de Minas Gerais, Brasil

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

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

4. Entanglement Witness [2]:
A Hermitian operator W with positive mean value for all separable states.
Tr{Wr}<0
r is entangled
The optimal entanglement witness Wopt to the state r is the one that maximises |Tr{Wr}|.
For the case of two qubits, we have [3]:
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)
[3] F. G. S. L. Brandão and R. O. Vianna, Int. J. Quant. Inf 4, 331 (2006)

5. Constructing the state
where
and
We have

6. 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)

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

8. Measuring RR(r)=-Tr{Wopt r)

12. Conclusions:
We have presented a method for probing the shape of the separable states’set. 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.