Linear entropy

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

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

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 ${\displaystyle \gamma \,}$ of a state by

${\displaystyle S_{L}\,=\,1-\gamma \,.}$

Motivation

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

${\displaystyle S\,{\dot {=}}\,-{\mbox{Tr}}(\rho \ln \rho )=-\langle \ln \rho \rangle \,.}$

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,

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

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

${\displaystyle S_{L}\,{\dot {=}}\,{\tfrac {d}{d-1}}(1-{\mbox{Tr}}(\rho ^{2}))\,,}$

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.