On symmetric SL-invariant polynomials in four qubits

Abstract

We find the generating set of SL-invariant polynomials in four qubits that are also invariant under permutations of the qubits. The set consists of four polynomials of degrees 2, 6, 8, and 12, for which we find an elegant expression in the space of critical states. These invariants are the degrees if the basic invariants of the invariants for F4, and in fact, the group plays an important role in this note. In addition, we show that the hyperdeterminant in four qubits is the only SL-invariant polynomial (up to powers of itself) that is non-vanishing precisely on the set of generic states.