Purity

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 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

10. %   requires: nothing

11. %   author: Nathaniel Johnston (nathaniel@njohnston.ca)

12. %   package: QETLAB

13. %   last updated: October 15, 2014

15. function gamma = Purity(rho)

17. gamma = real(trace(rho^2)); % "real" gets rid of close-to-0 imaginary part