IsotropicState

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IsotropicState
Produces an isotropic state

Other toolboxes required none
Related functions WernerState
Function category 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