Difference between revisions of "EntangledSubspace"
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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. | ||
Latest revision as of 01:42, 1 August 2023
| EntangledSubspace | |
| Produces a basis of an r-entangled subspace | |
| Other toolboxes required | none |
|---|---|
| Related functions | SchmidtRank |
| Function category | Entanglement and separability |
EntangledSubspace is a function that that creates a basis of an entangled subspace (i.e., a subspace of \(\mathbb{C}^m \otimes \mathbb{C}^n\) in which every pure state is entangled), or more generally a basis of an r-entangled subspace (i.e., a subspace of \(\mathbb{C}^m \otimes \mathbb{C}^n\) in which every pure state has Schmidt rank strictly greater than r).
Syntax
- E = EntangledSubspace(DIM,LOCALDIM)
- E = EntangledSubspace(DIM,LOCALDIM,R)
Argument descriptions
- DIM: The dimension of the desired subspace.
- LOCALDIM: 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.
- R (optional, default 1): 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 \(\mathbb{C}^3 \otimes \mathbb{C}^3\) has dimension 4. We can generate an entangled subspace of this dimension:
>> 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) 3The 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:
>> 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 3Indeed, 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 \(\mathbb{C}^4 \otimes \mathbb{C}^5\) has dimension 6. We can generate a 2-entangled subspace of this dimension as follows:
>> 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 0Notes
The largest r-entangled subspace of \(\mathbb{C}^m \otimes \mathbb{C}^n\) 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 [1]. The basis produced by this function is sparse.
Source code
Click here to view this function's source code on github.
References
- ↑ T. S. Cubitt, A. Montanaro, and A. Winter. On the dimension of subspaces with bounded Schmidt rank. J. Math. Phys. 49:022107, 2008. E-print: arXiv:0706.0705 [quant-ph]