Visualizing extremal positive maps in unital and trace-preserving form

Abstract

We define an entanglement witness in a composite quantum system as an observable having nonnegative expectation value in every separable state. Then a state is entangled if and only if it has a negative expectation value of some entanglement witness. Equivalent representations of entanglement witnesses are as nonnegative biquadratic forms or as positive linear maps of Hermitian matrices. As reported elsewhere, we have studied extremal entanglement witnesses in dimension $3\times 3$ by constructing numerical examples of generic extremal nonnegative forms. These are so complicated that we do not know how to handle them other than by numerical methods. However, the corresponding extremal positive maps can be presented graphically, as we attempt to do in the present paper. We understand that a positive map is extremal when the image of $\mathcal{D}$, the set of density matrices, fills out $\mathcal{D}$ maximally, in a certain sense. For the graphical presentation of a map we transform it to a standard form where it is unital and trace preserving. We present an iterative algorithm for the transformation, which converges rapidly in all our numerical examples and presumably works for any positive map. This standard form of an entanglement witness is unique up to unitary product transformations.