# Linear entropy

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

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 of a state by

## Motivation

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

The linear entropy then is obtained by expanding ln *ρ* = ln (1−(1−*ρ*)), around a pure state, *ρ*^{2}=*ρ*; that is, expanding in terms of the non-negative matrix 1−*ρ* in the formal Mercator series for the logarithm,

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

which ensures that the quantity ranges from zero to unity.

## References

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