Clapeyron's theorem (elasticity)

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 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

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