In quantum mechanics, and especially quantum information theory, the linear entropy or impurity of a state is a scalar defined as
\( S_L \, \dot= \, 1 - \mbox{Tr}(\rho^2) \, \)
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 \( \gamma \, \) 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
\( S \, \dot= \, -\mbox{Tr}(\rho \ln \rho) = -\langle \ln \rho \rangle \, . \)
The linear entropy then is obtained by expanding ln ρ = ln (1−(1−ρ)), around a pure state, ρ2=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.
Alternate definition
Some authors[1] define linear entropy with a different normalization
\( S_L \, \dot= \, \tfrac{d}{d-1} (1 - \mbox{Tr}(\rho^2) ) \, , \)
which ensures that the quantity ranges from zero to unity.
References
Nicholas A. Peters; Tzu-Chieh Wei; Paul G. Kwiat (2004). "Mixed state sensitivity of several quantum information benchmarks". Physical Review A. 70 (5): 052309. arXiv:quant-ph/0407172. Bibcode:2004PhRvA..70e2309P. doi:10.1103/PhysRevA.70.052309.
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