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Extremal entanglement witnesses

We study extremal entanglement witnesses on a bipartite quantum system. We define the cone of witnesses as the dual of the set of separable density matrices, thus $\textrm{Tr}\,Ωρ\geq 0$ when $Ω$ is a witness and $ρ$ a pure product state, $ρ=ψψ^{\dagger}$ with $ψ=ϕ\otimesχ$. The set of witnesses of unit trace is a compact convex set, defined by its extremal points. The expectation value $f(ϕ,χ)=\mathrm{Tr}\,Ωρ$ as a function of $ϕ$ and $χ$ is a nonnegative biquadratic form. Every zero of $f(ϕ,χ)$ imposes real-linear constraints on $f$ and $Ω$. The Hessian matrix at the zero must be nonnegative. Its eigenvectors with zero eigenvalue, if any, we call Hessian zeros. A zero of $f(ϕ,χ)$ is quadratic if it has no Hessian zeros, otherwise it is quartic. We call a witness quadratic if it has only quadratic zeros, and quartic otherwise. We prove that a witness is extremal if and only if no other witness has the same, or a larger, set of zeros and Hessian zeros. A quadratic extremal witness has a minimum number of isolated zeros depending on dimensions. If a witness is not extremal, the constraints defined by its zeros and Hessian zeros determine all directions in which to search for witness