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The linear entropy is a measure of mixedness in
quantum states. Its main feature is that it is easy to compute. It is a
scalar defined as
SL˙=1−Tr(ρ2)
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.)
Linear entropy is trivially related to the
purity
γ of a state by
SL = 1 − γ .
Motivation
The linear entropy is an approximation to the
Von
Neumann entropy S , which
is defined as
S˙=−Tr(ρlnρ).
The linear entropy is obtained by approximating ln ρ with the first order term
(ρ − 1) in the Mercator
series
−Tr(ρln ρ) → −Tr(ρ(ρ − 1)) = Tr(ρ − ρ2) = 1 − Tr(ρ2) = SL
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
SL˙=dd−1(1−Tr(ρ2)).
This ensures that the quantity ranges from zero to unity.
Category:
Entropy
Category:
Stubs
Last modified:
Monday, October 26, 2015 - 17:37
