I was reading http://arxiv.org/abs/quant-ph/0101012v4 and one of the axioms is that there needs to be a continuous reversible transformation between states. What is the difference between that and a continuous bijection.
Is it that a bijection is a function but a transformation is something more general?
In topology a continuous bijection which is reversibly continuous is a homeomorphism. It need not be the case that every continuous bijection is a homeomorphism. Consider any bijection between the discrete space $2^\omega$ and $\mathbb R$. It is continuous trivially, yet cannot be reversibly continuous, as a singleton is open in $2^\omega$ but not in $\mathbb R$. This of course shows that more generally a bijection need not be structure preserving.