The '''linear entropy''' is a measure of mixedness in quantum states. Its main feature is that it is easy to compute. It is a scalar defined as
:$S\_L\; \backslash ,\; \backslash dot=\; \backslash ,\; 1\; -\; \backslash mbox\{Tr\}(\backslash rho^2)\; \backslash ,$
where $\backslash rho\; \backslash ,$ is the density matrix of the state.
The linear entropy can range between zero, corresponding to a completely pure state, and $(1\; -\; 1/d)\; \backslash ,$, corresponding to a completely mixed state. (Here, $d\; \backslash ,$ is the dimension of the density matrix.)
Linear entropy is trivially related to the purity $\backslash gamma\; \backslash ,$ of a state by
:$S\_L\; \backslash ,\; =\; \backslash ,\; 1\; -\; \backslash gamma\; \backslash ,\; .$
==Motivation==
The linear entropy is an approximation to the Von Neumann entropy $S\backslash ,$, which is defined as
:$S\; \backslash ,\; \backslash dot=\; \backslash ,\; -\backslash mbox\{Tr\}(\backslash rho\; \backslash ln\; \backslash rho)\; \backslash ,\; .$
The linear entropy is obtained by approximating $\backslash ln\; \backslash rho\; \backslash ,$ with the first order term $(\backslash rho\; -\; 1)\; \backslash ,$ in the Mercator series
:$-\backslash mbox\{Tr\}(\backslash rho\; \backslash ln\; \backslash rho)\; \backslash ,\; \backslash to\; -\backslash mbox\{Tr\}(\backslash rho\; (\backslash rho-1))\; =\; \backslash mbox\{Tr\}(\backslash rho\; -\backslash rho^2)\; =\; 1\; -\; \backslash mbox\{Tr\}(\backslash rho^2)\; =\; S\_L$
where the unit trace property of the density matrix has been used to get the second to last equality.
The linear entropy and Von Neumann entropy are similar measures of the mixedness of a state, although the linear entropy is easier to calculate because it does not require the diagonalization of the density matrix.
==Alternate definition==
Some authors define linear entropy with a different normalization
:$S\_L\; \backslash ,\; \backslash dot=\; \backslash ,\; \backslash frac\{d\}\{d-1\}\; (1\; -\; \backslash mbox\{Tr\}(\backslash rho^2)\; )\; \backslash ,\; .$
This ensures that the quantity ranges from zero to unity.
Category: Entropy
Category: Stubs

## Last modified:

Monday, October 26, 2015 - 17:37