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The geometric measure of entanglement for symmetric states · Around

2009

The geometric measure of entanglement for symmetric states

Hübener, R., Kleinmann, M., Wei, T. -C. et al.

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Is the closest product state to a symmetric entangled multiparticle state also symmetric? This question has appeared in the recent literature concerning the geometric measure of entanglement.

First, we show that a positive answer can be derived from results concerning symmetric multilinear forms and homogeneous polynomials, implying that the closest product state can be chosen to be symmetric.
We then prove the stronger result that the closest product state to any symmetric multiparticle quantum state is necessarily symmetric.
Moreover, we discuss generalizations of our result and the case of translationally invariant states, which can occur in spin models.

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O. D. Kellogg, Math. Z. 27
1933
Earlier work this paper cites.
G. Pólya, Acta Math. 68
1937
Earlier work this paper cites.
L. Hörmander, Math. Scand. 2
1954
Earlier work this paper cites.
A. Grothendieck, Bol. Soc. Mat. São Paulo 8
1956
Earlier work this paper cites.
C.K. Majumdar and D.K. Ghosh, J. Math. Phys. 10
1969
Earlier work this paper cites.
A. Defant and K. Floret, Tensor Norms and Operator Ideals
1993
Earlier work this paper cites.
A well-known exception is the entanglement of formation (or the concurrence) for two qubits, see W.K. Wootters, Phys. Rev. Lett. 80
1998
Earlier work this paper cites.
S. Dineen, Complex Analysis on Infinite Dimensional Spaces , Springer, London-Berlin-Heidelberg, (1999)
1999
Earlier work this paper cites.

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A polynomial P P is N N -homogeneous if P ⁡ ( λ x ) = λ N P ( x ) . P(\lambda x)=\lambda^{N}P(x)

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Note that e e and e ∗ e^{*} are vectors with complex conjugated elements, i.e., e = e r + i e i e=e_{r}+ie_{i} and e ∗ = e r − i e i e^{*}=e_{r}-ie_{i} where e r , e i ∈ ℝ k e_{r},e_{i}\in\mathbbm{R}^{k} . Hence e r , e i e_{r},e_{i} is an example for a real basis, being linearly independent since e ≢ e ∗ e\not\equiv e^{*} . In fact, any real orthonormal basis of span ℂ ( { e r , e i } ) \mathrm{span}_{\mathbbm{C}}(\{e_{r},e_{i}\}) can be used as f 1 , f 2 f_{1},f_{2} , as long as they fulfill (iii)

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