Linear entropy

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

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

where $\rho \,$ is the density matrix of the state. The subscript $L\,$ is to distinguish this quantity from the Von Neumann entropy. 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.

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

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

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.