On symmetric SL-invariant polynomials in four qubits

  • Gilad GourEmail author
  • Nolan R. Wallach
Part of the Progress in Mathematics book series (PM, volume 257)


We find the generating set of \(\mathop{\mathrm{SL}}\nolimits\)-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 F 4, 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 \(\mathop{\mathrm{SL}}\nolimits\)-invariant polynomial (up to powers of itself) that is non-vanishing precisely on the set of generic states.


Quantum entanglement SL-invariant polynomials Four qubits Permutation invariant 

Mathematics Subject Classification

22 81 



GG research is supported by NSERC, NW was partially supported by an NSF Summer Grant


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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada
  2. 2.Department of MathematicsUniversity of California at San DiegoLa JollaUSA

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