Title: Four-qubit critical states

 

Abstract: Let H be the Hilbert space of unnormalized four-qubit state vectors. By a result of Verstraete, Dehaene, and De Moor, ||f||^(1/m) is an entanglement monotone for any complex homogeneous polynomial \(f\) on \(H\) of degree \(m > 0\) invariant under the action of the SLOCC group. We observe that many highly entangled or useful four-qubit states that appear in prior literature are stationary points of \(||f||\) for natural choices of \(f\). This motivates the search for more stationary points. Using the notion of critical points (in the sense of the Kempf-Ness theorem) together with results from Vinberg’s theory of theta groups, we reduce the complexity of the problem significantly. After reduction, we solve systems of polynomial equations with techniques from numerical algebraic geometry, recovering the stationary points known to us at the beginning and more.