ReductionMap

ReductionMap
Produces the reduction map
Other toolboxes required	none
Related functions	ChoiMap
Function category	Superoperators

ReductionMap is a function that returns the Choi matrix of the linear map that acts as follows:

\[\Phi(X) := \mathrm{Tr}(X)I - X,\]

where $I$ is the identity matrix. This map is positive.

Syntax

• R = ReductionMap(DIM)
• R = ReductionMap(DIM,K)

Argument descriptions

• DIM: The dimension of the reduction map. That is, the size of the matrices that the reduction map acts on.
• K (optional, default 1): If this positive integer is provided, the script will instead return the Choi matrix of the following linear map:

\[\Phi(X) := K\cdot\mathrm{Tr}(X)I - X.\]

Examples

The reduction map is positive

The following code returns the Choi matrix of the 3-dimensional reduction map and then verifies that the reduction map is indeed positive (i.e., verifies that its Choi matrix is block positive):

>> R = ReductionMap(3)
 
R =
 
   (5,1)       -1
   (9,1)       -1
   (2,2)        1
   (3,3)        1
   (4,4)        1
   (1,5)       -1
   (9,5)       -1
   (6,6)        1
   (7,7)        1
   (8,8)        1
   (1,9)       -1
   (5,9)       -1
 
>> IsBlockPositive(R) % verify that the reduction map is positive
 
ans =
 
     1

Higher values of K

It is known that the generalization of the reduction map that is provided by the optional argument K is always K-positive, but not (K+1)-positive. The following code verifies this in the K = 2 case:

>> R = ReductionMap(3,2)
 
R =
 
   (1,1)        1
   (5,1)       -1
   (9,1)       -1
   (2,2)        2
   (3,3)        2
   (4,4)        2
   (1,5)       -1
   (5,5)        1
   (9,5)       -1
   (6,6)        2
   (7,7)        2
   (8,8)        2
   (1,9)       -1
   (5,9)       -1
   (9,9)        1
 
>> IsBlockPositive(R,1) % verify that this map is positive
 
ans =
 
     1
 
>> IsBlockPositive(R,2) % verify that this map is 2-positive
 
ans =
 
     1
 
>> IsBlockPositive(R,3) % see that this map is not 3-positive
 
ans =
 
     0

Source code

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

1. %%  REDUCTIONMAP    Produces the reduction map

2. %   This function has one required argument:

3. %     DIM: a positive integer (the dimension of the reduction map)

4. %

5. %   R = ReductionMap(DIM) is the Choi matrix of the reduction map, which is

6. %   a positive map on DIM-by-DIM matrices.

7. %

8. %   This function has one optional argument:

9. %     K (default 1)

10. %   

11. %   R = ReductionMap(DIM,K) is the Choi matrix of the map defined by

12. %   R(X) = K*trace(X)*eye(DIM^2) - X. This map is K-positive.

13. %

14. %   URL: http://www.qetlab.com/ReductionMap

16. %   requires: iden.m, MaxEntangled.m, opt_args.m

17. %   author: Nathaniel Johnston (nathaniel@njohnston.ca)

18. %   package: QETLAB

19. %   last updated: September 29, 2014

21. function R = ReductionMap(dim,varargin)

23. % set optional argument defaults: k=1 (the usual reduction map)

24. [k] = opt_args({ 1 },varargin{:});

26. psi = MaxEntangled(dim,1,0);

27. R = k*speye(dim^2) - psi*psi';