IsProductVector - Revision history

Nathaniel at 15:13, 22 September 2014

2014-09-22T15:13:47Z

Nathaniel: Updated documentation using GeSHi

2014-09-21T20:03:32Z

Updated documentation using GeSHi

Nathaniel: /* A product state's decomposition */

2012-11-28T15:26:38Z

A product state's decomposition

Nathaniel: /* Examples */

2012-11-27T16:48:45Z

Examples

Nathaniel: Created page with "{{Function |name=IsProductVector |desc=Determines if a pure state is a product vector |req=opt_args
SchmidtDecomposition |rel=IsProductOperator
[..."

2012-11-26T21:32:00Z

Created page with "{{Function |name=IsProductVector |desc=Determines if a pure state is a product vector |req=opt_args<br />SchmidtDecomposition |rel=IsProductOperator<br />[..."

New page

{{Function
|name=IsProductVector
|desc=Determines if a [[pure state]] is a [[product vector]]
|req=[[opt_args]]<br />[[SchmidtDecomposition]]
|rel=[[IsProductOperator]]<br />[[SchmidtRank]]
|upd=November 26, 2012
|v=1.00}}
<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.

==Syntax==
* <tt>IPV = IsProductVector(VEC)</tt>
* <tt>IPV = IsProductVector(VEC,DIM)</tt>
* <tt>[IPV,DEC] = IsProductVector(VEC,DIM)</tt>

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

===Output arguments===
* <tt>IPV</tt>: Either 1 or 0, indicating that <tt>VEC</tt> is or is not a product vector.
* <tt>DEC</tt> (optional): If <tt>IPV = 1</tt> (i.e., <tt>VEC</tt> is a product vector), then <tt>DEC</tt> is a cell containing two or more vectors, the tensor product of which is <tt>VEC</tt>. If <tt>IPV = 0</tt> then <tt>DEC</tt> 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 $\mathbb{C}^2 \otimes \mathbb{C}^3 \otimes \mathbb{C}^5$:
<pre<noinclude></noinclude>>
>> v = [[RandomStateVector|RandomStateVector(30)]];
>> IsProductVector(v,[2,3,5])

ans =

0
</pre<noinclude></noinclude>>

@@ Line 2: / Line 2: @@
 |name=IsProductVector
 |desc=Determines if a [[pure state]] is a [[product vector]]
-|req=[[opt_args]]<br />[[SchmidtDecomposition]]
+|rel=[[IsProductOperator]]<br />[[SchmidtDecomposition]]<br />[[SchmidtRank]]
 |upd=November 26, 2012
 |v=1.00}}
@@ Line 27: / Line 26: @@
 ===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 $\mathbb{C}^2 \otimes \mathbb{C}^3 \otimes \mathbb{C}^5$:
-<pre<noinclude></noinclude>>
+<syntaxhighlight>
->> v = [[RandomStateVector|RandomStateVector(30)]];
+>> v = RandomStateVector(30);
 >> IsProductVector(v,[2,3,5])
@@ Line 34: / Line 33: @@
-</pre<noinclude></noinclude>>
+</syntaxhighlight>
 ===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.
-<pre<noinclude></noinclude>>
+<syntaxhighlight>
 >> v = [1 0 0 0 1 0 0 0]/sqrt(2);
 >> [ipv,dec] = IsProductVector(v,[2,2,2])
@@ Line 67: / Line 66: @@
->> [[Tensor|Tensor(dec)]]
+>> Tensor(dec)
 ans =
@@ Line 79: / Line 78: @@
-</pre<noinclude></noinclude>>
+</syntaxhighlight>

@@ Line 50: / Line 50: @@
      [2x1 double]    [2x1 double]    [2x1 double]
->> celldisp(dec)
+>> celldisp(dec) % display the contents of dec
 dec{1} =
@@ Line 67: / Line 67: @@
->> [[MultiKron|MultiKron(dec)]]
+>> [[Tensor|Tensor(dec)]]
 ans =