It is often said that e.g., in describing the collapse of states in Quantum Mechanics (QM), speaking or analyzing in terms of information provides a more solid footing compared to focusing on changes of coherence of the state, because coherence is a relative concept. Naturally then, extending the same idea, for mixed states it is easier to argue with loss of information, for instance if our system was initially in a pure state.
The aim of this post is to better understand what is meant by the relativity of coherence, and in what sense is arguing in terms of "amount of information" more absolute.
I came across these questions when thinking about non-unitarity of the collapse of a wavefunction, or in the other way around the non-unitarity of going from a mixed state to a pure one. Before asking myself these questions, I had always just assumed that in a sense there's a duality to concepts of coherence and information in QM, and now I'm very curious to find out why this may not be necessarily the case.
Please do feel free to give examples if you see them complementing well your arguments. For instance, are there trivial examples where by changing say the basis or something along those lines, one can show that our starting state has undergone a change in coherence, whereas the amount of information it contains/ed is left unchanged.
In order to describe the non-unitarity of the collapse of a wavefunction, in general we need the two following properties: unitary transformations preserve scalar product and in turn norms, and quantum measurements produce states that we call eigenstates characterized by the fact that they are not affected by repeated measurements (related: quantum Zeno paradox). With these two properties in mind about unitary transformations and quantum measurements, one can easily demonstrate why the collapse of a wavefunction cannot result from a unitary transformation, as contradictions will arise if one tries to do so.
The relevant point here is that, unitary transformations map pure states to pure states, whereas measurements, whether they are applied to pure or mixed states, map out to pure states only. One can rephrase these observations by saying: unitary operations are by definition reversible, whereas after a collapse: a) state coherence is lost (we say the system decohered) b) information on the original state is lost, either a) or b) is meant to say that the collapse is an irreversible process.
The question here at hand is basically asking: Are the statements a) and b) equivalent here? or is b) the more correct way of expressing the irreversibility of a collapse in QM? If we look at mixed states, again b) seems to work better as clearly information is lost. I hope this helps to better clarify the gist of the question.