# What is "linear" in linear entropy?

Why is the linear entropy, defined by $$S_L = 1 - \textrm{Tr} \rho^2$$, called linear?

$$S_L$$ called linear because it's obtained from the usual definition of von Neumann entropy $$S = -\mathrm{Tr}(\rho \ln \rho)$$ by taking a linear approximation for the natural log $$\ln \rho = \rho - 1$$. Then $$S_L = \mathrm{Tr}(\rho(1 - \rho)) = 1 - \mathrm{Tr}(\rho^2)$$. This simplification is often useful for calculations as you can avoid having to diagonalize the density matrix. For more details, have a look at Wikipedia.