# IsProductVector

 Other toolboxes required IsProductVector Determines if a pure state is a product vector none IsProductOperatorSchmidtDecompositionSchmidtRank Entanglement and separability

IsProductVector is a function that determines if a bipartite or multipartite vector (e.g., a pure quantum state) is a product vector or not. If it is a product vector, its tensor decomposition can be provided.

## Syntax

• IPV = IsProductVector(VEC)
• IPV = IsProductVector(VEC,DIM)
• [IPV,DEC] = IsProductVector(VEC,DIM)

## Argument descriptions

### Input arguments

• VEC: A vector that lives in a bipartite or multipartite Hilbert space.
• DIM (optional, by default has two subsystems of equal dimension): A specification of the dimensions of the subsystems that VEC lives in. DIM can be provided in one of two ways:
• If DIM is a scalar, it is assumed that VEC lives in the tensor product of two subsystems, the first of which has dimension DIM and the second of which has dimension length(VEC)/DIM.
• If $VEC \in \mathbb{C}^{n_1} \otimes \cdots \otimes \mathbb{C}^{n_p}$ then DIM should be a vector containing the dimensions of the subsystems (i.e., DIM = [n_1, ..., n_p]).

### Output arguments

• IPV: Either 1 or 0, indicating that VEC is or is not a product vector.
• DEC (optional): If IPV = 1 (i.e., VEC is a product vector), then DEC is a cell containing two or more vectors, the tensor product of which is VEC. If IPV = 0 then DEC is meaningless.

## Examples

### A random example

A randomly-generated pure state will almost surely not be a product vector. The following code demonstrates this for a random pure state chosen from $\mathbb{C}^2 \otimes \mathbb{C}^3 \otimes \mathbb{C}^5$:

>> v = RandomStateVector(30);
>> IsProductVector(v,[2,3,5])

ans =

0

### A product state's decomposition

The following code determines that a certain pure state living in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ is a product state, and provides a decomposition of that product state. It is then verified that the tensor product of the vectors in that decomposition do indeed give the original state.

>> v = [1 0 0 0 1 0 0 0]/sqrt(2);
>> [ipv,dec] = IsProductVector(v,[2,2,2])

ipv =

1

dec =

[2x1 double]    [2x1 double]    [2x1 double]

>> celldisp(dec) % display the contents of dec

dec{1} =

0.7071
0.7071

dec{2} =

1.0000
0

dec{3} =

1
0

>> Tensor(dec)

ans =

0.7071
0
0
0
0.7071
0
0
0

## Source code

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

1. %%  ISPRODUCTVECTOR   Determines if a pure state is a product vector
2. %   This function has one required argument:
3. %     VEC: a vector living in the tensor product of two or more subsystems
4. %
5. %   IPV = IsProductVector(VEC) is either 1 or 0, indicating that VEC is or
6. %   is not a product state (note that VEC is assumed to be bipartite unless
7. %   the optional argument DIM (see below) is specified).
8. %
9. %   This function has one optional input argument:
10. %     DIM (default has two subsystems of equal dimension)
11. %
12. %   [IPV,DEC] = IsProductVector(VEC,DIM) indicates that VEC is or is not a
13. %   product state, as above. DIM is a vector containing the dimensions of
14. %   the subsystems that VEC lives on. If IPV = 1 then DEC is a product
15. %   decomposition of VEC. More specifically, DEC is a cell containing
16. %   vectors whose tensor product equals VEC.
17. %
18. %   URL: http://www.qetlab.com/IsProductVector
19. 
20. %   requires: opt_args.m, SchmidtDecomposition.m
21. %   author: Nathaniel Johnston (nathaniel@njohnston.ca)
22. %   package: QETLAB
23. %   last updated: November 26, 2012
24. 
25. function [ipv,dec] = IsProductVector(vec,varargin)
26. 
27. lv = length(vec);
28. 
29. % set optional argument defaults: dim=sqrt(length(vec))
30. [dim] = opt_args({ round(sqrt(lv)) },varargin{:});
31. 
32. % allow the user to enter a single number for dim
33. num_sys = length(dim);
34. if(num_sys == 1)
35.     dim = [dim,lv/dim];
36.     if abs(dim(2) - round(dim(2))) >= 2*lv*eps
37.         error('IsProductVector:InvalidDim','The value of DIM must evenly divide length(VEC); please provide a DIM array containing the dimensions of the subsystems.');
38.     end
39.     dim(2) = round(dim(2));
40.     num_sys = 2;
41. end
42. 
43. % if there are only two subsystems, just use the Schmidt decomposition
44. if(num_sys == 2)
45.     [s,u,v] = SchmidtDecomposition(vec,dim,2);
46.     ipv = (s(2) <= prod(dim) * eps(s(1)));
47.     if(ipv) % provide this even if not requested, since it is needed if this function was called as part of its recursive algorithm (see below)
48.         u = u*sqrt(s(1));
49.         v = v*sqrt(s(1));
50.         dec = {u(:,1) v(:,1)};
51.     end
52. 
53. % if there are more subsystems, recursively use the Schmidt decomposition across many cuts until we are sure
54. else
55.     [ipv,dec] = IsProductVector(vec,[dim(1)*dim(2),dim(3:end)]);
56.     if(ipv)
57.         [ipv,tdec] = IsProductVector(dec{1},[dim(1),dim(2)]);
58.         if(ipv)
59.             dec = [tdec dec{2:end}];
60.         end
61.     end
62. end
63. 
64. if(~ipv)
65.     dec = 0;
66. end