# Tensor

 Other toolboxes required Tensor Kronecker tensor product of two or more matrices none TensorSum Basic operation

Tensor is a function that produces the Kronecker (tensor) product of two or more matrices, and thus extends MATLAB's built-in kron function.

## Syntax

• KRN = Tensor(A)
• KRN = Tensor(A,M)
• KRN = Tensor(A,B,C,...)

## Argument descriptions

• A: If A is a cell, then KRN is the Kronecker product of all matrices within A.
• M (optional): A scalar indicating how many times A should be tensored with itself (if M is provided, A must be a matrix).
• B,C,... (optional): Matrices. If these are provided, then A must be a matrix, and the output is $A \otimes B \otimes C \otimes \cdots$.

## Examples

### Several different input formats

The Tensor function accepts input in many different formats, so that you may use whichever is most convenient at a particular time. For example, the following three code snippets all result in the same operator: Tensor(A,3), Tensor(A,A,A), and Tensor({A,A,A}).

### Multiple copies of Werner states

When investigating the NPPT bound entanglement conjecture, you may want to tensor Werner states with themselves multiple times. The following code tensors a particular Werner state with itself 6 times:

>> rho = WernerState(3,1/2,1);
>> rho6 = Tensor(rho,6);

Note that rho6 is a 531441-by-531441 matrix, so we don't display it here.

## Source code

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

1. %%  TENSOR   Kronecker tensor product of two or more matrices or vectors
2. %   This function can be called in one of three ways:
3. %
4. %   KRN = Tensor(A,B,C,...) returns the Kronecker product A \otimes B
5. %   \otimes C \otimes ...
6. %
7. %   KRN = Tensor(A,m) returns the Kronecker product of A with itself m
8. %   times.
9. %
10. %   KRN = Tensor(A), if A is a cell, returns the Kronecker product of
11. %   all matrices contained within A.
12. %
13. %   URL: http://www.qetlab.com/Tensor
14. 
15. %   requires: nothing
16. %   author: Nathaniel Johnston (nathaniel@njohnston.ca)
17. %   package: QETLAB
18. %   last updated: May 10, 2016
19. 
20. function krn = Tensor(A,varargin)
21. 
22. % If A is a cell, tensor together each element of the cell.
23. if(iscell(A))
24.     krn = A{1};
25.     for j = 2:length(A);
26.         krn = kron(krn,A{j});
27.     end
28. 
29. % If two arguments were received and the second is a scalar, tensor A with itself that many times.
30. elseif(nargin == 2 && length(varargin{1}) == 1)
31.     % Tensor naively if we only want a few copies.
32.     if(varargin{1} <= 4 || (length(A) > 3 && ~issparse(A)))
33.         krn = 1;
34.         for j = 1:varargin{1};
35.             krn = kron(krn,A);
36.         end
37. 
38.     % Be more clever (a-la exponentiation by squaring) if we want lots of
39.     % copies. For n copies, this procedure only does O(log(n)) Kronecker
40.     % products. Doesn't help much for full matrices, but in practice is
41.     % about a 50% speedup for some sparse matrices.
42.     else
43.         l2 = floor(log2(varargin{1}));
44.         krn_cell = cell(1,l2+1);
45.         b = de2bi(varargin{1});
46. 
47.         krn = 1;
48.         krn_cell{1} = A;
49.         for j = 1:l2
50.             krn_cell{j+1} = kron(krn_cell{j},krn_cell{j});
51.             if(b(j) == 1)
52.                 krn = kron(krn,krn_cell{j});
53.             end
54.         end
55.         krn = kron(krn,krn_cell{l2+1});
56.     end
57. 
58. % If two or more arguments were received and the second isn't a scalar, tensor together all arguments.
59. else
60.     krn = A;
61.     for j = 2:nargin;
62.         krn = kron(krn,varargin{j-1});
63.     end
64. end