Four-qubit critical states
Luke Oeding, Ian Tan
TL;DR
The paper develops a Vinberg-theoretic framework to enumerate stationary points of symmetric SLOCC-invariant entanglement measures for four qubits, reducing the search to a 4-dimensional Cartan subspace and exploiting Weyl symmetry. Analytic results are obtained for the $|\mathcal{F}_1|$ case, while numerical nonlinear algebra computes stationary points for $|\mathcal{F}_3|$ and $|\mathcal{F}_4|$, yielding an extended set of representative states that include all critical states from prior surveys. Several stationary points lead to five- and six-qubit absolutely maximally entangled states and corresponding pure quantum error correcting codes via a reverse Rains construction, illustrating a tangible application in quantum information. The work deepens the connection between Kempf-Ness theory, Vinberg theory, and quantum entanglement, showing that stationary points can reveal useful entangled states and actionable code designs beyond local maxima.
Abstract
Verstraete, Dehaene, and De Moor (2003) showed that SLOCC invariants provide entanglement monotones. We observe that many highly entangled or useful four-qubit states that appear in prior literature are stationary points of such entanglement measures. This motivates the search for more stationary points. We use the notion of critical points (in the sense of the Kempf-Ness theorem) together with Vinberg theory to reduce the complexity of the problem significantly. We solve the corresponding systems utilizing modern numerical nonlinear algebra methods and reduce the solutions by natural symmetries. This method produces an extended list of four-qubit stationary points, which includes all the critical states in the survey by Enriquez et al (2016). To illustrate the potential for application, we discuss the use of these states to generate pure five-qubit and six-qubit quantum error correcting codes by reversing a construction of Rains (1996).