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Linear entropy : ウィキペディア英語版
Linear entropy
In quantum mechanics, and especially quantum information theory, the linear entropy or impurity of a state is a scalar defined as
:S_L \, \dot= \, 1 - \mbox(\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 \gamma \, of a state by
:S_L \, = \, 1 - \gamma \, .
==Motivation==

The linear entropy is a lower approximation to the (quantum) von Neumann entropy ''S'', which is defined as
:S \, \dot= \, -\mbox(\rho \ln \rho) = -\langle \ln \rho \rangle \, .
The linear entropy then is obtained by expanding ln ''ρ'' = ln (1−(1−''ρ'')), around a pure state, ''ρ''2=''ρ''; that is, expanding in terms of the non-negative matrix 1−''ρ'' in the formal Mercator series for the logarithm,
: - \langle \ln \rho \rangle = \langle 1- \rho \rangle + \langle (1- \rho )^2 \rangle/2 + \langle (1- \rho)^3 \rangle /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.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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