Geometrically Induced Singular Behavior of Entanglement

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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
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Random Robustness 1
Quantifying Entanglement
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)
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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)).
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Entanglement Witness 2
A Hermitian operator W with positive mean value
for all separable states.
TrWrlt0
r is entangled
The optimal entanglement witness Wopt to the
state r is the one that maximises TrWr.
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)
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Constructing the state
where
and
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, 042328
(2006)
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Experimental Setup
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5 P. G. Kwiat et. al., Phys. Rev. Lett. 75,
4337 (1995)
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Measuring RR(r)-TrWopt r)
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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.