MaxEntangled

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MaxEntangled
Produces a maximally entangled bipartite pure state

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

MaxEntangled is a function that returns the canonical maximally entangled bipartite pure state. The state can be chosen to be either full or sparse.

Syntax

  • PSI = MaxEntangled(DIM)
  • PSI = MaxEntangled(DIM,SP)
  • PSI = MaxEntangled(DIM,SP,NRML)

Argument descriptions

  • DIM: The dimension of the local subsystems on which PSI lives.
  • SP (optional, default 0): A flag (either 1 or 0) indicating that PSI should or should not be sparse.
  • NRML (optional, default 1): A flag (either 1 or 0) indicating that PSI should or should not be scaled to have Euclidean norm 1. If NRML=0 then PSI has Euclidean norm sqrt(DIM) and every element of PSI is 0 or 1.

Examples

A maximally entangled qubit state

To generate a maximally entangled pair of qubits you can use the following line of code:

>> MaxEntangled(2)
 
ans =
 
    0.7071
         0
         0
    0.7071

If you want an unnormalized version of this state in which each entry of the vector is 0 or 1, specify NRML=0:

>> MaxEntangled(2,0,0)
 
ans =
 
     1
     0
     0
     1

In larger systems

When DIM is large, it is usually best to specify SP=1 in order to save memory. For example, this code produces a maximally entangled pure state in $\mathbb{C}^{10} \otimes \mathbb{C}^{10}$:

>> MaxEntangled(10,1)
 
ans =
 
   (1,1)       0.3162
  (12,1)       0.3162
  (23,1)       0.3162
  (34,1)       0.3162
  (45,1)       0.3162
  (56,1)       0.3162
  (67,1)       0.3162
  (78,1)       0.3162
  (89,1)       0.3162
 (100,1)       0.3162

Source code

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

  1. %%  MAXENTANGLED    Produces a maximally entangled bipartite pure state
  2. %   This function has one required argument:
  3. %     DIM: the local dimension
  4. %
  5. %   PSI = MaxEntangled(DIM) is a DIM^2-by-1 column vector representing the
  6. %   standard bipartite maximally entangled pure state \sum_i \ket{ii}
  7. %   (normalized to have norm 1).
  8. %
  9. %   This function has two optional arguments:
  10. %     SP (default 0)
  11. %     NRML (default 1)
  12. %   
  13. %   PSI = MaxEntangled(DIM,SP,NRML) produces a maximally entangled pure
  14. %   state as above that is sparse if SP = 1 and is full if SP = 0. The pure
  15. %   state is normalized to have Euclidean norm 1 if NRML = 1, and it is
  16. %   unnormalized (i.e., each entry in the vector is 0 or 1 and the
  17. %   Euclidean norm of the vector is sqrt(DIM)) if NRML = 0.
  18. %
  19. %   URL: http://www.qetlab.com/MaxEntangled
  20.  
  21. %   requires: iden.m, opt_args.m
  22. %   author: Nathaniel Johnston (nathaniel@njohnston.ca)
  23. %   package: QETLAB
  24. %   last updated: November 28, 2012
  25.  
  26. function psi = MaxEntangled(dim,varargin)
  27.  
  28. % set optional argument defaults: sp=0, nrml=1
  29. [sp,nrml] = opt_args({ 0, 1 },varargin{:});
  30.  
  31. % construct the vector
  32. psi = reshape(iden(dim,sp),dim^2,1);
  33. if(nrml)
  34.     psi = psi/sqrt(dim);
  35. end