Nathaniel at 01:42, 1 August 2023

2023-08-01T01:42:38Z

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	===Notes===		==Notes==
	The largest r-entangled subspace of <math>\mathbb{C}^m \otimes \mathbb{C}^n</math> has dimension (m-r)(n-r), so requesting a larger subspace will produce an error message. The method of construction that this function uses was described in <ref name="CMW">T. S. Cubitt, A. Montanaro, and A. Winter. On the dimension of subspaces with bounded Schmidt rank. <em>J. Math. Phys.</em> 49:022107, 2008. E-print: [https://arxiv.org/abs/0706.0705 arXiv:0706.0705] [quant-ph]</ref>. The basis produced by this function is sparse.		The largest r-entangled subspace of <math>\mathbb{C}^m \otimes \mathbb{C}^n</math> has dimension (m-r)(n-r), so requesting a larger subspace will produce an error message. The method of construction that this function uses was described in <ref name="CMW">T. S. Cubitt, A. Montanaro, and A. Winter. On the dimension of subspaces with bounded Schmidt rank. <em>J. Math. Phys.</em> 49:022107, 2008. E-print: [https://arxiv.org/abs/0706.0705 arXiv:0706.0705] [quant-ph]</ref>. The basis produced by this function is sparse.

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{{Function
|name=EntangledSubspace
|desc=Produces a basis of an r-entangled subspace
|rel=[[SchmidtRank]]
|cat=[[List of functions#Entanglement_and_separability|Entanglement and separability]]
|upd=May 6, 2022
|v=1.00}}
<tt>'''EntangledSubspace'''</tt> is a [[List of functions|function]] that that creates a basis of an entangled subspace (i.e., a subspace of <math>\mathbb{C}^m \otimes \mathbb{C}^n</math> in which every pure state is entangled), or more generally a basis of an r-entangled subspace (i.e., a subspace of <math>\mathbb{C}^m \otimes \mathbb{C}^n</math> in which every pure state has [[Schmidt rank]] strictly greater than r).

==Syntax==
* <tt>E = EntangledSubspace(DIM,LOCALDIM)</tt>
* <tt>E = EntangledSubspace(DIM,LOCALDIM,R)</tt>

==Argument descriptions==
* <tt>DIM</tt>: The dimension of the desired subspace.
* <tt>LOCALDIM</tt>: A scalar indicating the local dimension of the (bipartite) ambient space that the subspace will live in, or a 2-entry vector indicating its two local dimensions.
* <tt>R</tt> (optional, default <tt>1</tt>): A (strict) lower bound on the Schmidt rank of pure states in the subspace.

==Examples==
===A two-qutrit entangled subspace of maximum dimension===
The largest entangled subspace of <math>\mathbb{C}^3 \otimes \mathbb{C}^3</math> has dimension 4. We can generate an entangled subspace of this dimension:
<syntaxhighlight>
>> E = EntangledSubspace(4,3)

E =

(2,1) 1
(6,1) 1
(1,2) 1
(5,2) 1
(9,2) 1
(4,3) 1
(8,3) 1
(1,4) 1
(5,4) 2
(9,4) 3
</syntaxhighlight>
The columns of this matrix are the basis vectors of the subspace. To get a clearer picture of this, it is perhaps useful to convert the function's sparse output to full:
<syntaxhighlight>
>> full(E)

ans =

0 1 0 1
1 0 0 0
0 0 0 0
0 0 1 0
0 1 0 2
1 0 0 0
0 0 0 0
0 0 1 0
0 1 0 3
</syntaxhighlight>
Indeed, those four columns are linearly independent, and any linear combination of them is entangled.

===A 2-entangled subspace with unequal local dimensions===
The largest 2-entangled subspace of <math>\mathbb{C}^4 \otimes \mathbb{C}^5</math> has dimension 6. We can generate a 2-entangled subspace of this dimension as follows:
<syntaxhighlight>
>> E = full(EntangledSubspace(6,[4,5],2))

E =

0 0 1 0 0 1
0 1 0 0 1 0
1 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 1 0 0
0 0 1 0 0 2
0 1 0 0 2 0
1 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 1 0 0
0 0 1 0 0 3
0 1 0 0 3 0
1 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 1 0 0
0 0 1 0 0 4
0 1 0 0 4 0
</syntaxhighlight>

===Notes===
The largest r-entangled subspace of <math>\mathbb{C}^m \otimes \mathbb{C}^n</math> has dimension (m-r)(n-r), so requesting a larger subspace will produce an error message. The method of construction that this function uses was described in <ref name="CMW">T. S. Cubitt, A. Montanaro, and A. Winter. On the dimension of subspaces with bounded Schmidt rank. <em>J. Math. Phys.</em> 49:022107, 2008. E-print: [https://arxiv.org/abs/0706.0705 arXiv:0706.0705] [quant-ph]</ref>. The basis produced by this function is sparse.

{{SourceCode|name=EntangledSubspace}}

==References==
<references />

EntangledSubspace - Revision history

Nathaniel at 01:42, 1 August 2023

Nathaniel: Created page