# Purity

 Other toolboxes required Purity Computes the purity of a quantum state none Miscellaneous

Purity is a function that computes the purity of a quantum state $\rho$ (i.e., it computes the quantity ${\rm Tr}(\rho^2)$).

## Syntax

• GAMMA = Purity(RHO)

## Argument descriptions

• RHO: A density matrix to have its purity computed.

## Examples

### Purity of pure states

Pure states have purity equal to 1, as illustrated by the following code:

>> phi = RandomStateVector(3);
>> Purity(phi*phi')

ans =

1.0000

>> Purity(RandomDensityMatrix(3,0,1))

ans =

1.0000

### Purity of mixed states

If $\rho \in M_d$ is mixed then its purity is strictly less than 1. Its purity attains its minimum value of $1/d$ if and only if $\rho$ is the maximally-mixed state (i.e., the scaled identity operator).

>> Purity(WernerState(2,1/4)) % the state WernerState(2,1/4) acts on 4-dimensional space

ans =

0.2653

>> Purity(eye(4)/4)

ans =

0.2500