  It is a well-established fact in quantum information theory, that uniform
averaging over the collective action of a unitary group on a multipartite
quantum state projects the state to a form equivalent to a permutation operator
of the subsystems. Hence states equivalent to permutation operators are
untouched by collective unitary noise. A trivial observation shows that uniform
averaging over permutation operators projects the state into a form with
block-diagonal structure equivalent to the one of the collective action of the
unitary group. We introduce a name for this property: duality of averaging. The
mathematical reason behind this duality is the fact that the collective action
of the unitary group on the tensor product state space of a multipartite
quantum system and the action of the permutation operations are mutual
commutants when treated as matrix algebras. Such pairs of matrix algebras are
known as dual reductive pairs. In this work we show, that in the case of finite
dimensional quantum systems such duality of averaging holds for any pairs of
symmetry groups being dual reductive pairs, regardless of whether they are
compact or not, as long as the averaging operation is defined via iterated
integral over the Cartan decomposition of the group action. Although our result
is very general, we focus much attention on the concrete example of a dual
reductive pair consisting of collective action of special linear matrices and
permutation operations, which physically corresponds to averaging multipartite
quantum states over non-unitary SLOCC-type (Stochastic Local Operations and
Classical Communication) operations. In this context we show, that noiseless
subsystems known from collective unitary averaging persist in the case of SLOCC
averaging in a conditional way: upon postselection to specific invariant
subspaces.
