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, Spain2Imperial College London, England3Departamento de Física, Universidade Federal de Minas Gerais, Brasil4Departamento de Matemática, Universidade Federal de Minas Gerais, Brasil

RR() = Minimum s such that the state below is separable:

RR 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)

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 (q) describe an analytic curve in the space of states, then RR((q)) is also a Cm function.

Singularities in the border of the set of separable states will show up as singularities in RR((q)).

A Hermitian operator W with positive mean value for all separable states.

The optimal entanglement witness Wopt 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)

Constructing the state

where and

We have

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)

Experimental Setup:

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

[5]

Measuring RR()=-Tr{Wopt)

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.