On polynomial invariants of several qubits

  • D. Ž. Ðoković, A. Osterloh
  • Journal of Mathematical Physics, March 2009, American Institute of Physics
  • DOI: 10.1063/1.3075830

Craftswork and tricks in SL-invariant entanglement buisiness

What is it about?

We summarizes the main results for fourqubit invariants and establish the correspondence between two different sets of SL-invariants. Next, we give a new and complete characterization for the space of five-qubit SL-invariants for degrees 8, 10, and 12. We also make a few remarks on the invariants of degrees 14 and 16 that give room for speculations on the completeness of this characterization as far as general degrees are concerned. All the work is performed by using the omega process by Cayley and the local SL-invariant operatores from earlier work. We determine the Hilbert series for SL*-invariants and present an interesting connection between the Cayley omega-process, spinor metric, and local invariant operators. We furthermore demonstrate that a third SL-independent local operator completes the set for two qubits.

Why is it important?

This work impressively shows how strong is the impact of local SL-invariant operators when tayloring some invariant that vanishes on certain (which might be all) bipartitions. The craftswork is continued until there is a hint towards a presumably complete characterization for five qubits. Also the dictionary for translating certain invariants in different will become helpful to better understand the matters underneath for qubits.

Perspectives

Dr. Andreas Osterloh

This work has some practical part, where we address the feasibility of making a complete characterization of SL-invariants beyond 4 qubits. The technique to write the invariant operators in terms of certain permutations from arXiv:0810.1240) has delivered the third invariant operator and it is a glimpse how one could systematically create such operators for higher dimensions.

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http://dx.doi.org/10.1063/1.3075830

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