BreuerState

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BreuerState
Produces a Breuer state of even dimension ≥ 2

Other toolboxes required none
Related functions ChessboardState
Function category Special states, vectors, and operators

BreuerState is a function that produces a two-qudit "Breuer state" (i.e., the state defined in [1]). These states are of interest because they are bound entangled. The output of this function is sparse.

Syntax

  • BREUER_STATE = BreuerState(DIM,LAMBDA)

Argument descriptions

  • DIM: The local dimension (must be ≥ 2 and even).
  • LAMBDA: The weight of the singlet component in the state (see [1] for details). A positive real number between 0 and 1. The state will be separable if and only if LAMBDA = 0 and it will have positive partial transpose if and only if LAMBDA <= 1/(DIM + 2)).

Examples

A 4 ⊗ 4 bound entangled state

The following code generates a bound entangled Breuer state in $M_4 \otimes M_4$ and then verifies that has positive partial transpose and is entangled (and is thus bound entangled):

>> rho = full(BreuerState(4,0.1));
>> IsPPT(rho)
 
ans =
 
     1
 
>> IsSeparable(rho)
Determined to be entangled by not having a 2-copy PPT symmetric extension. Reference:
A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri. A complete family of separability criteria. Phys. Rev. A, 69:022308, 2004.
 
ans =
 
     0

Source code

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

  1. %%	BREUERSTATE    Produces a Breuer state of even dimension >= 2
  2. %	This function has two required arguments:
  3. %     DIM: the local dimension (for dim >= 2 and even) 
  4. %     LAMBDA: describes the weight of the singlet component
  5. %
  6. %	BREUER_STATE = BreuerState(DIM,LAMBDA) gives a Breuer bound entangled 
  7. %   state for two qudits of local dimension DIM, with the LAMBDA parameter
  8. %   describing the weight of the singlet component, as described in [1].
  9. %   The Breuer state that is returned is sparse.
  10. %
  11. %	References:
  12. %   [1] H-P. Breuer. Optimal entanglement criterion for mixed quantum 
  13. %       states. E-print: arXiv:quant-ph/0605036, 2006.
  14. %
  15. %	URL: http://www.qetlab.com/BreuerState
  16.  
  17. %	requires: MaxEntangled.m, SymmetricProjection.m
  18. % 	authors: Vincent Russo (vrusso@uwaterloo.ca)
  19. %            Nathaniel Johnston (nathaniel@njohnston.ca)
  20. %	package: QETLAB 
  21. %	last updated: December 15, 2014
  22.  
  23. function breuer_state = BreuerState(dim, lambda)
  24.  
  25. % Make sure that the dimension is even.
  26. if mod(dim,2) == 1 || dim <= 0
  27.     error('BreuerState:InvalidDim','DIM must be an even positive integer.');
  28. end
  29.  
  30. % Start by generating a specific maximally-entangled state PSI.
  31. V = fliplr(sparse(diag((-1).^mod(1:dim,2))));
  32. psi = kron(speye(dim),V)*MaxEntangled(dim,1);
  33.  
  34. % Mix the maximally-entangled state PSI with the normalized projector to
  35. % the symmetric subspace.
  36. breuer_state = lambda * (psi*psi') + (1-lambda) * 2*SymmetricProjection(dim)/(dim*(dim+1));

References

  1. 1.0 1.1 H.-P. Breuer. Optimal entanglement criterion for mixed quantum states. Phys. Rev. Lett., 97:080501, 2006. E-print: arXiv:quant-ph/0605036