# RandomSuperoperator

 Other toolboxes required RandomSuperoperator Generates a random superoperator (completely positive map) none RandomDensityMatrixRandomPOVMRandomStateVectorRandomUnitary Random things

RandomSuperoperator is a function that generates a random completely positive map satisfying some user-specified properties (such as being trace-preserving, being unital, and/or having a specified number of Kraus operators).

## Syntax

• PHI = RandomSuperoperator(DIM)
• PHI = RandomSuperoperator(DIM,TP)
• PHI = RandomSuperoperator(DIM,TP,UN)
• PHI = RandomSuperoperator(DIM,TP,UN,RE)
• PHI = RandomSuperoperator(DIM,TP,UN,RE,KR)

## Argument descriptions

• DIM: Either a scalar, indicating that PHI should act on DIM-by-DIM matrices, or a 1-by-2 vector, indicating that PHI should take DIM(1)-by-DIM(1) matrices as input and return DIM(2)-by-DIM(2) matrices as output.
• TP (optional, default 1): A flag (either 1 or 0) indicating that PHI should or should not be trace-preserving.
• UN (optional, default 0): A flag (either 1 or 0) indicating that PHI should or should not be unital (i.e., satisfy $\Phi(I) = I$).
• RE (optional, default 0): A flag (either 1 or 0) indicating that the Choi matrix of PHI (equivalently, its Kraus operators) should only have real entries (RE = 1) or that it is allowed to have complex entries (RE = 0).
• KR (optional, default prod(DIM)): The maximal number of Kraus operators of the superoperator to be produced. With probability 1, it will have exactly KR Kraus operators (if KR ≤ prod(DIM)).

## Examples

### Random qubit channel

The following code generates a random qubit channel and verifies that it is indeed a qubit channel:

>> Phi = RandomSuperoperator(2);
>> ApplyMap(eye(2),DualMap(Phi)) % the dual map is unital if and only if Phi is trace-preserving

ans =

1.0000 - 0.0000i  -0.0000 - 0.0000i
-0.0000 + 0.0000i   1.0000 - 0.0000i

>> IsCP(Phi) % verify that Phi is completely positive

ans =

1

### Random unital channel with specified number of Kraus operators

You can request completely positive maps with any combination of the optional arguments as constraints. For example, the following code requests a random unital quantum channel that sends 3-by-3 matrices to 4-by-4 matrices and has 5 Kraus operators. Note that a warning is produced because, strictly speaking, no such map exists (it's impossible for a map to be both trace-preserving and unital unless the input and output dimensions are the same). Instead, the map is that is produced is trace-preserving and sends the identity matrix to a scalar multiple of the identity matrix.

>> Phi = RandomSuperoperator([3,4],1,1,0,5);
Warning: There does not exist a unital, trace-preserving map in the case when the input and output
dimensions are unequal. The identity matrix will map to a *multiple* of the identity matrix.
> In RandomSuperoperator at 45

>> ApplyMap(eye(3),Phi) % verify that Phi is (up to scaling) unital

ans =

0.7500 + 0.0000i  -0.0000 - 0.0000i  -0.0000 - 0.0000i  -0.0000 - 0.0000i
-0.0000 + 0.0000i   0.7500 + 0.0000i   0.0000 + 0.0000i  -0.0000 - 0.0000i
-0.0000 + 0.0000i   0.0000 - 0.0000i   0.7500 + 0.0000i  -0.0000 + 0.0000i
-0.0000 + 0.0000i  -0.0000 + 0.0000i  -0.0000 - 0.0000i   0.7500 + 0.0000i

>> ApplyMap(eye(4),DualMap(Phi,[3,4])) % verify that Phi is trace-preserving

ans =

1.0000 - 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
0.0000 - 0.0000i   1.0000 - 0.0000i   0.0000 + 0.0000i
0.0000 - 0.0000i   0.0000 - 0.0000i   1.0000 - 0.0000i

>> IsCP(Phi) % verify that Phi is completely positive

ans =

1

>> KrausOperators(Phi,[3,4]) % verify that Phi has 5 Kraus operators

ans =

[4x3 double]
[4x3 double]
[4x3 double]
[4x3 double]
[4x3 double]

## Notes

The superoperator is not generated with regard to any well-known or well-studied distribution, but it can output any superoperator satisfying the specified constraints (unital, trace-preserving, number of Kraus operators), and it derives from uniform spherical measure in a fairly straightforward way.

## Source code

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

1. %%  RANDOMSUPEROPERATOR    Generates a random superoperator
2. %   This function has one required argument:
3. %     DIM: either a scalar or a 1-by-2 vector specifying the input and
4. %          output dimensions of the superoperator, in that order
5. %
6. %   PHI = RandomSuperoperator(DIM) generates the Choi matrix of a random
7. %   quantum channel (i.e., a completely positive, trace-preserving linear
8. %   map) acting on DIM-by-DIM matrices.
9. %
10. %   This function has four optional arguments:
11. %     TP (default 1)
12. %     UN (default 0)
13. %     RE (default 0)
14. %     KR (default prod(DIM))
15. %
16. %   PHI = RandomSuperoperator(DIM,TP,UN,RE,KR) generates the Choi matrix of
17. %   a random superoperator that is trace-preserving if TP=1, is unital if
18. %   UN=1, has all real entries if RE=1, and has KR or fewer Kraus operators
19. %   (it will have exactly KR Kraus operators with probability 1).
20. %
21. %   URL: http://www.qetlab.com/RandomSuperoperator
22. 
23. %   requires: iden.m, MaxEntangled.m, OperatorSinkhorn.m, opt_args.m,
24. %             PartialTrace.m, PermuteSystems.m, RandomDensityMatrix.m,
25. %             RandomStateVector.m, RandomUnitary.m, SchmidtDecomposition.m,
26. %             Swap.m
27. %
28. %   author: Nathaniel Johnston (nathaniel@njohnston.ca)
29. %   package: QETLAB
30. %   last updated: September 30, 2014
31. 
32. function Phi = RandomSuperoperator(dim,varargin)
33. 
34. % allow the user to enter a single number for dim
35. if(length(dim) == 1)
36.     dim = [dim,dim];
37. end
38. pd = prod(dim);
39. 
40. % set optional argument defaults: tp=1, un=0, re=0, kr=prod(dim)
41. [tp,un,re,kr] = opt_args({ 1, 0, 0, pd },varargin{:});
42. 
43. if(tp == 1 && un == 1 && dim(1) ~= dim(2))
44.     warning('RandomSuperoperator:InvalidDim','There does not exist a unital, trace-preserving map in the case when the input and output dimensions are unequal. The identity matrix will map to a *multiple* of the identity matrix.');
45. end
46. 
47. % There is a probability 0 chance that the operator Sinkhorn iteration will
48. % get cranky. Thus we repeatedly try until we *don't* get an error
49. % (honestly, I'm being slightly overly cautious).
50. sing_err = 1;
51. while sing_err
52.     sing_err = 0;
53.     try
54.         % Generate the Choi matrix of a superoperator that is not
55.         % necessarily trace-preserving or unital. We will enforce those
56.         % conditions in a moment.
57.         Phi = RandomDensityMatrix(pd,re,kr,'haar');
58. 
59.         % Now set the appropriate partial traces to the identity.
60.         if(tp == 1 && un == 0)
61.             PT = kron(sqrtm(inv(PartialTrace(Phi,2,dim))),eye(dim(2)));
62.             Phi = PT*Phi*PT;
63.         elseif(tp == 0 && un == 1)
64.             PT = kron(eye(dim(1)),sqrtm(inv(PartialTrace(Phi,1,dim))));
65.             Phi = PT*Phi*PT;
66.         elseif(tp == 1 && un == 1)
67.             Phi = OperatorSinkhorn(Phi,dim)*dim(1);
68.         end
69.     catch err
70.         % Operator Sinkhorn error? Try, try again!
71.         if(strcmpi(err.identifier,'OperatorSinkhorn:LowRank'))
72.             sing_err = 1;
73.         else
74.             rethrow(err);
75.         end
76.     end
77. end