Linear entropy

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

$S_L \, \dot= \, 1 - \mbox{Tr}(\rho^2) \,$

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

1. 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.