Four-qubit critical states

TL;DR

This paper identifies and classifies critical four-qubit states using advanced algebraic methods, expanding the known set of such states and exploring their applications in quantum error correction.

Contribution

It introduces a novel algebraic approach combining Kempf-Ness and Vinberg theory to find all critical four-qubit states, including those previously known and new ones.

Findings

Extended list of four-qubit stationary states

Includes all states from prior surveys

Potential applications in quantum error correction

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).