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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?

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    $\begingroup$ Did you get to page 15 of the paper? You will find out that the author uses a very unusual notion of a "continuous transformation". In brief, it is an element of a compact connected Lie group acting linearly on a vector space. $\endgroup$ – Moishe Kohan Jun 30 '14 at 1:01
  • $\begingroup$ Not yet, I am still working my way through. Thanks for pointing out that this is a bit unusual. $\endgroup$ – Tim Swast Jul 1 '14 at 2:40
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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.

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  • $\begingroup$ Very true, but not relevant in the context of the question. $\endgroup$ – Moishe Kohan Jun 30 '14 at 1:02

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