Entropy

The entropy $\bar{S}$ of an ensemble can be defined in terms of the density operator $\hat{\rho}$ as follows,

$$S \equiv -k Tr\{\hat{\rho} \hat{\rho} \},$$ (22)

where $k$ is the Botzmann constant. Equation (22) is the Von Neumann definition of entropy. This is the most fundamental definition of $S$ because it is given in terms of the density operator $\hat{\rho}$, which provides the most complete description of an ensemble. In particular, the Gibbs entropy formula,

$$S = -k \sum_k p_k p_k,$$ (23)

can be obtained from Eq. (22) by substituting $\hat{\rho}$ in accord with Eq. (11). From Eq. (23) one can see that the entropy of a pure state is zero, while the entropy of a statistical mixture is always positive. Therefore,

$$S \geq 0,$$ (24)

which is the fourth law of Thermodynamics.