Difference between revisions of "IsProductVector"
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|name=IsProductVector | |name=IsProductVector | ||
|desc=Determines if a [[pure state]] is a [[product vector]] | |desc=Determines if a [[pure state]] is a [[product vector]] | ||
| − | | | + | |rel=[[IsProductOperator]]<br />[[SchmidtDecomposition]]<br />[[SchmidtRank]] |
| − | + | |cat=[[List of functions#Entanglement_and_separability|Entanglement and separability]] | |
|upd=November 26, 2012 | |upd=November 26, 2012 | ||
| − | |v= | + | |v=0.50}} |
<tt>'''IsProductVector'''</tt> is a [[List of functions|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. | <tt>'''IsProductVector'''</tt> is a [[List of functions|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. | ||
| Line 27: | Line 27: | ||
===A random example=== | ===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 $\mathbb{C}^2 \otimes \mathbb{C}^3 \otimes \mathbb{C}^5$: | 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 $\mathbb{C}^2 \otimes \mathbb{C}^3 \otimes \mathbb{C}^5$: | ||
| − | < | + | <syntaxhighlight> |
| − | >> v = | + | >> v = RandomStateVector(30); |
>> IsProductVector(v,[2,3,5]) | >> IsProductVector(v,[2,3,5]) | ||
| Line 34: | Line 34: | ||
0 | 0 | ||
| − | </ | + | </syntaxhighlight> |
===A product state's decomposition=== | ===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. | 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. | ||
| − | < | + | <syntaxhighlight> |
>> v = [1 0 0 0 1 0 0 0]/sqrt(2); | >> v = [1 0 0 0 1 0 0 0]/sqrt(2); | ||
>> [ipv,dec] = IsProductVector(v,[2,2,2]) | >> [ipv,dec] = IsProductVector(v,[2,2,2]) | ||
| Line 50: | Line 50: | ||
[2x1 double] [2x1 double] [2x1 double] | [2x1 double] [2x1 double] [2x1 double] | ||
| − | >> celldisp(dec) | + | >> celldisp(dec) % display the contents of dec |
dec{1} = | dec{1} = | ||
| Line 67: | Line 67: | ||
0 | 0 | ||
| − | >> | + | >> Tensor(dec) |
ans = | ans = | ||
| Line 79: | Line 79: | ||
0 | 0 | ||
0 | 0 | ||
| − | </ | + | </syntaxhighlight> |
| + | |||
| + | {{SourceCode|name=IsProductVector}} | ||
Latest revision as of 15:13, 22 September 2014
| IsProductVector | |
| Determines if a pure state is a product vector | |
| Other toolboxes required | none |
|---|---|
| Related functions | IsProductOperator SchmidtDecomposition SchmidtRank |
| Function category | 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 =
0A 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
0Source code
Click here to view this function's source code on github.