# IsotropicState

 Other toolboxes required IsotropicState Produces an isotropic state none WernerState Special states, vectors, and operators

IsotropicState is a function that returns an isotropic state (i.e., a state of the following form):[1]

$$\displaystyle\rho_\alpha := \frac{1-\alpha}{d^2}I\otimes I + \alpha|\psi_+\rangle\langle\psi_+| \in M_d \otimes M_d,$$

where $|\psi_+\rangle:=\frac{1}{\sqrt{d}}\sum_j|j\rangle\otimes|j\rangle$ is the standard maximally-entangled pure state. Note that the output of this function is a sparse matrix.

## Syntax

• RHO = IsotropicState(DIM,ALPHA)

## Argument descriptions

• DIM: Dimension of the local subsystems on which RHO acts.
• ALPHA: A parameter that specifies which isotropic state is to be returned. In particular, RHO = (1-ALPHA)*I/DIM^2 + ALPHA*E, where I is the identity operator and E is the projection onto the standard maximally-entangled pure state on two copies of DIM-dimensional space. In order for RHO to be positive semidefinite (and hence a valid density matrix), it must be the case that -1/(DIM^2-1) ≤ ALPHA ≤ 1.

## Examples

### A qutrit isotropic state

To generate the isotropic state with parameter $\alpha = 1/2$, the following code suffices:

>> full(IsotropicState(3,1/2))

ans =

0.2222         0         0         0    0.1667         0         0         0    0.1667
0    0.0556         0         0         0         0         0         0         0
0         0    0.0556         0         0         0         0         0         0
0         0         0    0.0556         0         0         0         0         0
0.1667         0         0         0    0.2222         0         0         0    0.1667
0         0         0         0         0    0.0556         0         0         0
0         0         0         0         0         0    0.0556         0         0
0         0         0         0         0         0         0    0.0556         0
0.1667         0         0         0    0.1667         0         0         0    0.2222

Isotropic states in general have a lot of zero entries, so this function always returns a sparse matrix. If you want a full matrix (as above), use MATLAB's full function.

>> IsotropicState(3,1/2)

ans =

(1,1)       0.2222
(5,1)       0.1667
(9,1)       0.1667
(2,2)       0.0556
(3,3)       0.0556
(4,4)       0.0556
(1,5)       0.1667
(5,5)       0.2222
(9,5)       0.1667
(6,6)       0.0556
(7,7)       0.0556
(8,8)       0.0556
(1,9)       0.1667
(5,9)       0.1667
(9,9)       0.2222

## Source code

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

1. %%  ISOTROPICSTATE    Produces an isotropic state
2. %   This function has two required arguments:
3. %     DIM: the local dimension
4. %     ALPHA: the parameter of the isotropic state
5. %
6. %   RHO = IsotropicState(DIM,ALPHA) is the isotropic state with parameter
7. %   ALPHA acting on (DIM*DIM)-dimensional space. More specifically, RHO is
8. %   the density operator defined by (1-ALPHA)*I/DIM^2 + ALPHA*E, where I is
9. %   the identity operator and E is the projection onto the standard
10. %   maximally-entangled pure state on two copies of DIM-dimensional space.
11. %
12. %   URL: http://www.qetlab.com/IsotropicState
13. 
14. %   requires: iden.m, MaxEntangled.m, opt_args.m
15. %   author: Nathaniel Johnston (nathaniel@njohnston.ca)
16. %   package: QETLAB
17. %   last updated: September 22, 2014
18. 
19. function rho = IsotropicState(dim,alpha)
20. 
21. % compute the isotropic state
22. psi = MaxEntangled(dim,1,0);
23. rho = (1-alpha)*speye(dim^2)/dim^2 + alpha*psi*psi'/dim;

## References

1. M. Horodecki and P. Horodecki. Reduction criterion of separability and limits for a class of distillation protocols. Phys. Rev. A, 59:4206–4216, 1999. E-print: arXiv:quant-ph/9708015