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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Using random matrix techniques and the theory of Matrix Product States we show that reduced density matrices of quantum spin chains have generically maximum entropy. arXiv:1201.6324v1 [quant-ph]

It is well known that quantum correlations for bipartite dichotomic measurements are those of the form γ = ( u i , v j ) n i,j=1 , where the vectors u i and v j are in the unit ball of a real Hilbert space. In this work we study the probability of the nonlocal nature of these correlations as a function of α = m n , where the previous vectors are sampled according to the Haar measure in the unit sphere of R m . In particular, we prove the existence of an α 0 > 0 such that if α ≤ α 0 , γ is nonlocal with probability tending to 1 as n → ∞, while for α > 2, γ is local with probability tending to 1 as n → ∞.