# Purity

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Purity | |

Computes the purity of a quantum state | |

Other toolboxes required | none |
---|---|

Function category | 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
```

## Source code

Click here to view this function's source code on github.