In a big macroscopic system this is imposible in practice, because you only get a parcial knowledge of a few important properties, not of every detail.
Moreover, in most systems it is imposible to know everything about the system, but in some specially build systems like a quantum computer you can know the state of all the qbits.
If they can read a bit now that had itself set even one microsecond in the future, flash traders will arrive with truckloads of money.
If someone downvotes this can they reply to explain why?
Anyhow, both formulations of entropy - the thermodynamical one with the integral and the statistical physics one with multiplicity - are equivalent. At least the latter already contains statistics because of the combinatorial part.
And then there is one formulation of the 2nd law which states that on statistical average the entropy rises. So it doesn't break the law if it falls down. Given that this quantum computer doesn't involve many objects, the number of combinations isn't that high. But it's nice to observe Entropy falling! (And there are other situations in which this can happen anyway.)
'"This is one in a series of papers on the possibility of violating the second law of thermodynamics. That law is closely related to the notion of the arrow of time that posits the one-way direction of time from the past to the future," said the study's lead author Gordey Lesovik, who heads the Laboratory of the Physics of Quantum Information Technology at MIPT.'
When I don't see how violating the second law could even be a possibility.
We disagree in that this time reversal operation need energy. It's possible to do this experiment with a frequency doubler [1] crystal in reverse to split a photon in two photons with less energy, then use mirrors to reverse the photons, and then you will be able to "see" that the photons return to the crystal and produce the original photon. [Good luck aligning all the optical equipment perfectly. This is theoretically possible, but it would be very difficult to make the experiment. Perhaps it's easier with other particles.] Anyway, the time reversal operation in this experiment only use a few perfectly aligned mirrors, so it doesn't need additional energy.
In the experiment in the article, they use a setup that is like a quantum computer. It use energy to keep everything working, but the additional energy is not necessary for the main part of the experiment. (The energy is important to make the experiment possible, i.e. transform "perfect alignment" into "we can build this".)
[1] https://en.wikipedia.org/wiki/Optical_frequency_multiplier