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Separability and entanglement: what symmetries and geometry can say

Helena Braga, Simone Souza and Salomon S. Mizrahi

Departamento de Física, CCET, Universidade Federal de São Carlos

11th ICSSUR/ 4th Feynman Festival , June 22- 26 2009,

Palacký University, Olomouc – Czech Republic

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1. Introduction2. Polarization vector, correlation matrix3. Peres-Horodecki criterion4. New parameters and symmetries 5. Phase space: entangled-like and separable-like regions 6. Distance, concurrence and negativity7. Examples8. Conclusions

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Paradigm of entanglement: the two-qubit system

Bohm, two spin-1/2 particles: 1952

Lee and Yang, intrinsic parity Phys. Rev. 104, 822 (1956)

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Multipartite state up to 2nd order correlation: 4 X 4 matrix (15

parameters)

General multipartite state

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In matrix form

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Separability

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Reduction to seven parameters and the X-from

Transposition and Peres-Horodecki criterion

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Additional symmetries related to the eigenvalues of the transposed matrix

The eigenvalues of the TM can be obtained from

the original state by choosing six among different local reflections

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Introducing new parameters

Eigenvalues of the original matrix

Cond. of pos.

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Do the same with the transposed or locally reflected matrix

According to PHC if both ‘distances’ of the locally reflectedmatrix are positive the state is separableotherwise it is entangled

Define ‘phase spaces’: X

X = 0 define conic surfaces

= 0 and

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Phase space can be divided in two regions, one for the entangled-like states and the other for the separable-like Systems having the X-form are met in the literarature,

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Werner statePeres-Horod state

A. Peres, PRL 77, 1413 (1996)Horodeckis, PLA 223, 1 (1996)

R.F. Werner, PRA 40, 4277 (1989)

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N. Gisin, Phys. Lett. A 210, 151 (1996)

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M.P. Almeida et al., Science 316, 579 (2007)

Experimental work, entangled photon polarization and simulating dynamical evolution

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S. Das and G.S. Agarwal, arXiv:0901.2114v2 [quant-ph] 20 May 2009

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Comparisons between entanglements measures

1. concurrence,

2. Vidal-Werner negativity

3. distance-negativity

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A. Peres, PRL 77, 1413 (1996)Horodeckis, PLA 223, 1 (1996)

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R.F. Werner, PRA 40, 4277 (1989)

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N. Gisin, Phys. Lett. A 210, 151 (1996)

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M.P.Almeida et al., Science 316, 579 (2007)

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S. Das and G.S. Agarwal, arXiv:0901.2114v2 [quant-ph] 20 May 2009

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Conclusions

1. Matrix form symmetries imply physical symmetries when expressed in terms of meaningful physical quantities.

2. Defining new parameters, a phase space and distances, the PHC acquires a geometrical meaning: one can follow the trajectories of states and to discern graphically between entangled and separable states.

3. The negative of squared distance (the distance negativity) can be used as a measure of entanglement, which is comparable to other measures.

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Thank you!