Linear entropy

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

${\displaystyle S_L\dot{=}1-\text{Tr}({\rho }^2)}$

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 ${\displaystyle \gamma }$ of a state by

${\displaystyle S_L=1-\gamma .}$

The linear entropy is a lower approximation to the (quantum) von Neumann entropy S, which is defined as

${\displaystyle S\dot{=}-\text{Tr}(\rho \ln \rho )=-⟨\ln \rho ⟩.}$

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,

${\displaystyle -⟨\ln \rho ⟩=⟨1-\rho ⟩+⟨(1-\rho {)}^2⟩/2+⟨(1-\rho {)}^3⟩/3+...,}$

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^[1] define linear entropy with a different normalization

${\displaystyle S_L\dot{=}\frac{d}{d-1}(1-\text{Tr}({\rho }^2)),}$

which ensures that the quantity ranges from zero to unity.

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