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That is, something which is in a superposition of obeying the exclusion principle and not. And if you have a collection of them, how will you write their wavefunction to be in a superposition of being symmetric and anti-symmetric?

I'm aware that recently people have made multi-species mixtures of ultracold atoms, of which some are fermions and some bosons. Do these mixtures count as superpositions?

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    $\begingroup$ Particles with different statistics are in different superselection sectors, so a state that has both fermions and bosons is a convex combination of the two. $\endgroup$
    – Phoenix87
    Aug 21, 2016 at 18:01
  • $\begingroup$ This is in some ways what supersymmetry does. $\endgroup$
    – Lawrence B. Crowell
    Aug 21, 2016 at 18:14
  • $\begingroup$ In particle physics you have the Lamda particle. It decays into a proton and a pion. so nature superposes them . en.wikipedia.org/wiki/Lambda_baryon $\endgroup$
    – anna v
    Aug 22, 2016 at 3:24

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As I explain in this answer, a superselection rule prevents a relative phase of bosons and fermions to be detectable, as the space of states decomposes into two pieces whose time evolution and observables don't have anything to do with each other.

In particular, observables behave for the "superposition" just as they would behave if you just had the boson and the fermion state separately as a statistical mixture. There is no meaning to superpositions from states from different superselection sectors since the usual interference terms that make such superpositions detectable always vanish.

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