EntangledSubspace

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 <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

• 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 <math>\mathbb{C}^3 \otimes \mathbb{C}^3</math> 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)        3

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:

>> 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

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:

>> 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

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 . The basis produced by this function is sparse.

Source code

Click here to view this function's source code on github.