Restrictions on defining microstates (Entropy) If we have an isolated system $Sb$ with thermodynamic entropy $Eb=X$
(and growing by the 2nd law of thermodynamics),
we could define an abstract system $Sa$ (containing the system $Sb$) but define the possible microstates as a single bit, being 1 (one) when $Eb>=X$, and 0 (zero) when $Eb < X$.
The microstate will tend to be $1$, so the entropy of the system $Sa$  tends to zero.

po-->0
p1-->1
Eb = - k [p0 * log (p0) + p1 * log (p1)] --> 0

How could it be possible that an isolated system have a decreasing entropy?
Does the growth (or lack of growth) of the entropy depend on the microstates choosen?
Finally, if there is a restriction on how to choose the definition of the microstates, then what is that restriction?
Thanks.
 A: I think the question you're really trying to ask is: what makes a set of microstates work for thermodynamics. In quantum mechanics, for a set of microstates, you can just take a set of orthogonal states of the system. For classical mechanics, you need to take a set of microstates which all have equal volume. Of course, what equal volume means depends on the measure you put on the system. This generally isn't covered in popular texts. (I don't know why; maybe because people who are popularizing physics think that measures are stupid, unnecessary, and overly complicated mathematical notions.) For thermodynamics of ideal gasses, you can choose a microstate to be one which contains equal volume in terms of the x and p (position and momentum) variables of the particles. If you chose an alternate measure, where the states had equal volumes in terms of x2 and p2 variables, I believe that thermodynamics would not work with this measure. 
For other kinds of systems, if you know enough physics, it is fairly easy to see what the proper generalization of the variables x and p to these system (hint: position and momentum are complementary variables). I don't know what the proper way to explain this is, though.
A: The entropy $E_b$ is a macroscopic thermodynamic quantity. Therefore, any quantity derived from it will also be a macroscopic quantity. Therefore, your states are not microstates. 
The general restriction for a microstates is that once someone tells you what microstate a system is in, you know "everything" there is to know about the system. In classical mechanics, this means that a microstate specifies all positions and momenta of all particles in the system. In quantum mechanics, it means that you specify the complete many-body wavefunction or many-body state.
Your curious state does not tell you what state your system is in: There are many ways the system can be  in state $1$, and many ways it can be in state $0$. Because of that, it is not a microstate.
So, the general restriction is that a microstate has a unique realization in the system.
