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

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


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, ρ2=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[edit]

Some authors[1] define linear entropy with a different normalization

which ensures that the quantity ranges from zero to unity.


  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/0407172Freely accessible. Bibcode:2004PhRvA..70e2309P. doi:10.1103/PhysRevA.70.052309.