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

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$S_L$ 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.

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