For an electron in its rest frame, we have an entropy
$$
S = \log 2,
$$
which comes from the 2 possible spin directions along z-axis.
If the measurement $S_z$ changes its state to $\left| + \right>$, the entropy goes to zero.
Does this violate the second law of thermodynamics?
Best Answer
The entropy of the measured system decreases, at the expense of the entropy of the detector, which goes up. The total entropy balance is positive (as irreversibly fixing detection results usually costs more entropy than is gained from the reduction in the measured system.
Thus the second law is not violated.
Edit: About deriving such results: In a fully microscopic description the entropy remains constant, but nothing can be measured as there is no microscopic concept of a permanent record of measurement results. One, therefore, needs appropriate coarse-graining assumptions, which take the place of Boltzmann's 19th century Stosszahl ansatz for classical molecular systems.
Coarse graining means that the density matrix is restricted to take a form depending only on macroscopically measurable parameters, and deviations from this form (due to the exact dynamics) are swept under the coarse-graining carpet. This leads to approximate macroscopic dynamics. The resulting description is dissipative in the Markovian limit: The entropy $Tr(-\rho\log\rho)$ strictly increases with time unless the system is already in equilibrium.
A book covering this nicely and in full detail for a number of coarse-graining recipes is Grabert's ''Projection operator techniques in nonequilibrium statistical mechanics''. For a readable summary of the basic technique, see, e.g., https://arxiv.org/abs/cond-mat/9612129.