  De Finetti theorems show how sufficiently exchangeable states are
well-approximated by convex combinations of i.i.d. states. Recently, it was
shown that in many quantum information applications a more relaxed de Finetti
reduction (i.e. only a matrix inequality between the symmetric state and one of
de Finetti form) is enough, and that it leads to more concise and elegant
arguments. Here we show several uses and general flexible applicability of a
constrained de Finetti reduction in quantum information theory, which was
recently discovered by Duan, Severini and Winter. In particular we show that
the technique can accommodate other symmetries commuting with the permutation
action, and permutation-invariant linear constraints. We then demonstrate that,
in some cases, it is also fruitful with convex constraints, in particular
separability in a bipartite setting. This is a constraint particularly
interesting in the context of the complexity class $\mathrm{QMA}(2)$ of
interactive quantum Merlin-Arthur games with unentangled provers, and our
results relate to the soundness gap amplification of $\mathrm{QMA}(2)$
protocols by parallel repetition. It is also relevant for the regularization of
certain entropic channel parameters. Finally, we explore an extension to
infinite-dimensional systems, which usually pose inherent problems to de
Finetti techniques in the quantum case.
