One of the less understood questions, and very fascinating at the same time, is the deep link that appears to exist between quantum and statistical mechanics. This can be summed up into the relation between the density matrix and the unitary evolution operator, that is
where the arrow stands for the variable change that connects temperature and time. Anyhow, we know that the thermodynamic limit, when applied to the density matrix, gives back standard thermodynamic. What if the same is done on the unitary evolution operator? We have given a general answer here. This paper is my contribution to the proceedings of DICE 2006 and was a talk that resumed all my work about this matter in this latest years. The answer is
In the limit of an increasingly large number of particles a quantum system becomes classical.
Notwithstanding a lot of evidence accumulated about, nobody did a proper experimental evaluation properly aimed at this and so we still stand without a clear understanding of the link between statistical and quantum mechanics. This will give a serious answer to the matter. It should be understood that a proper preparation of the quantum system can maintain it as long as we want in a quantum state. So, our conclusion above does not preclude at all quantum computation.
We just note that experimental evidence come out by Pastawski and his group (e.g. see here). The results of this group were striking and blatantly proved our conclusion above. Anyhow, their work met skepticism of the community impeding publication on relevant archival journals. This happened notwithstanding the evident goodness of the experimental work these people accomplished.
Our hope by now is that some brave researcher will try to exploit from an experimental point of view the possibility to understand in depth the deep link between quantum and statistical mechanics. I think the pay-off may be surely rewarding and could sign another milestone in the understanding of the quantum-classical border.