# Purity (quantum mechanics)

In quantum mechanics, and especially quantum information theory, the purity of a quantum state is a scalar defined as

$\gamma \, \equiv \, \mbox{Tr}(\rho^2) \,$

where $\rho \,$ is the density matrix of the state. The purity can range between unity, corresponding to a completely pure state, and $1/d \,$, corresponding to a completely mixed state. (Here, $d \,$ is the dimension of the density matrix.)

Purity is trivially related to the Linear entropy $S_L \,$ of a state by

$\gamma = 1-S_L \, .$

## Projectivity of a measurement

For a quantum measurement, the projectivity[1] is the purity of its pre-measurement state. This pre-measurement state is the main tool of the retrodictive approach of quantum physics, in which we make predictions about state preparations leading to a given measurement result. It allows us to determine in which kind of states the measured system was prepared for leading to such a result.

## References

1. ^ Taoufik Amri, Quantum behavior of measurement apparatus, arXiv:1001.3032 (2010).