Linear entropy

In quantum mechanics, and especially quantum information theory, the linear entropy of a state is a scalar defined as

$S_{L}\,{\dot {=}}\,1-{\mbox{Tr}}(\rho ^{2})\,$

where $\rho \,$ is the density matrix of the state.

The linear entropy can range between zero, corresponding to a completely pure state, and $(1-1/d)\,$, corresponding to a completely mixed state. (Here, $d\,$ is the dimension of the density matrix.)

Linear entropy is trivially related to the purity $\gamma \,$ of a state by

$S_{L}\,=\,1-\gamma \,.$

Motivation

The linear entropy is an approximation to the Von Neumann entropy $S\,$, which is defined as

$S\,{\dot {=}}\,-{\mbox{Tr}}(\rho \log _{2}\rho )\,.$

The linear entropy is obtained by approximating $\ln \rho \,$ with the first order term $(\rho -1)\,$ in the Mercator series

$-{\mbox{Tr}}(\rho \log _{2}\rho )\,\to -{\mbox{Tr}}(\rho (\rho -1))={\mbox{Tr}}(\rho -\rho ^{2})=1-{\mbox{Tr}}(\rho ^{2})=S_{L}$

where the unit trace property of the density matrix has been used to get the second to last equality.

The linear entropy and Von Neumann entropy are similar measures of the "mixedness" of a state, although the linear entropy is easier to calculate because it does not require the diagonalization of the density matrix.

Alternate definition

Some authors define linear entropy with a different normalization

$S_{L}\,{\dot {=}}\,{\frac {d}{d-1}}(1-{\mbox{Tr}}(\rho ^{2}))\,.$

This ensures that the quantity ranges from zero to unity.