where ρ 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.)
The linear entropy is trivially related to the purity of a state by
The linear entropy is a lower approximation to the (quantum) von Neumann entropy S, which is defined as
The linear entropy then is obtained by expanding ln ρ = ln (1−(1−ρ)), around a pure state, ρ2=ρ; that is, expanding in terms of the non-negative matrix 1−ρ in the formal Mercator series for the logarithm,
and retaining just the leading term.
The linear entropy and von Neumann entropy are similar measures of the degree of mixing of a state, although the linear entropy is easier to calculate, as it does not require diagonalization of the density matrix.
Some authors define linear entropy with a different normalization
which ensures that the quantity ranges from zero to unity.