On Symmetric SL-Invariant Polynomials in Four Qubits

TL;DR

This paper identifies the fundamental symmetric SL-invariant polynomials for four qubits, revealing their degrees, explicit forms, and the unique role of the Hyperdeterminant in characterizing generic states.

Contribution

It provides the complete generating set of symmetric SL-invariant polynomials for four qubits and characterizes the Hyperdeterminant as the unique non-vanishing invariant on generic states.

Findings

Generated set of four SL-invariant polynomials of degrees 2, 6, 8, and 12.

Explicit expressions for these polynomials in the space of critical states.

Hyperdeterminant is the only invariant non-vanishing on generic states.

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. 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.