BreuerState

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 ). 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 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

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

23. function breuer_state = BreuerState(dim, lambda)

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

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);

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));