Samiksha Jaiswal (Editor)

Linear entropy

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

Contents

S L = ˙ 1 − Tr ( ρ 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 γ of a state by

S L = 1 − γ .

Motivation

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

S = ˙ − Tr ( ρ ln ⁡ 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 define linear entropy with a different normalization

S L = ˙ d d − 1 ( 1 − Tr ( ρ 2 ) ) ,

which ensures that the quantity ranges from zero to unity.

References

Linear entropy Wikipedia
(Text) CC BY-SA