Physics:Linear entropy

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

[math]\displaystyle{ S_L \, \dot= \, 1 - \mbox{Tr}(\rho^2) \, }[/math]

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 [math]\displaystyle{ \gamma \, }[/math] of a state by

[math]\displaystyle{ S_L \, = \, 1 - \gamma \, . }[/math]

Motivation

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

[math]\displaystyle{ S \, \dot= \, -\mbox{Tr}(\rho \ln \rho) = -\langle \ln \rho \rangle \, . }[/math]

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,

[math]\displaystyle{ - \langle \ln \rho \rangle = \langle 1- \rho \rangle + \langle (1- \rho )^2 \rangle/2 + \langle (1- \rho)^3 \rangle /3 + ... ~, }[/math]

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

[math]\displaystyle{ S_L \, \dot= \, \tfrac{d}{d-1} (1 - \mbox{Tr}(\rho^2) ) \, , }[/math]

which ensures that the quantity ranges from zero to unity.