# 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

## Source code

Click on "expand" to the right to view the MATLAB source code for this function.

1. %%  PURITY    Computes the purity of a quantum state
2. %   This function has one required argument:
3. %     RHO: a density matrix
4. %
5. %   GAMMA = Purity(RHO) is the purity of the quantum state RHO (i.e., GAMMA
6. %   is the quantity trace(RHO^2)).
7. %
8. %   URL: http://www.qetlab.com/Purity
9. 
10. %   requires: nothing
11. %   author: Nathaniel Johnston (nathaniel@njohnston.ca)
12. %   package: QETLAB
13. %   last updated: October 15, 2014
14. 
15. function gamma = Purity(rho)
16. 
17. gamma = real(trace(rho^2)); % "real" gets rid of close-to-0 imaginary part