The most important lesson I learned from the reconstructions of
quantum mechanics is how fundamental reversibility is for the theory:
none of the reconstructions succeeded without postulating that the
dynamics of quantum states is reversible. Here we apply the same idea
to process matrices, by introducing a definition of what does it mean
for a process - with indefinite causal order - to be reversible. We
then show that there exist processes which cannot be purified into a
reversible process, and thus are probably not physical.
The reversible process which introduced can then be shown to have a
natural representation in terms of post-selected closed timelike
curves (P-CTCs). This representation, in its turn, gives a simple
scheme capable of simulating arbitrary processes via post-selection,
which is however clearly not optimal.