Linear entropy

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

SL=˙1−Tr(ρ2)

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 γ of a state by

SL=1−γ.

## Motivation

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

S=˙−Tr(ρln⁡ρ)=−⟨ln⁡ρ⟩.

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,

−⟨ln⁡ρ⟩=⟨1−ρ⟩+⟨(1−ρ)2⟩/2+⟨(1−ρ)3⟩/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

SL=˙dd−1(1−Tr(ρ2)),

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. doi:10.1103/PhysRevA.70.052309. Bibcode: 2004PhRvA..70e2309P.

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