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.