Difference between revisions of "ChoiMap"
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(Created page with "{{Function |name=ChoiMap |desc=Produces the Choi map or one of its generalizations |req=iden<br />MaxEntangled<br />opt_args |upd=August 5, 2013 |v=1.00}} <tt>'''C...") |
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: <math>\begin{bmatrix}x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \\ x_{31} & x_{32} & x_{33}\end{bmatrix} \mapsto \begin{bmatrix}ax_{11}+bx_{22}+cx_{33} & -x_{12} & -x_{13} \\ -x_{21} & cx_{11}+ax_{22}+bx_{33} & -x_{23} \\ -x_{31} & -x_{32} & bx_{11}+cx_{22}+ax_{33}\end{bmatrix},</math> | : <math>\begin{bmatrix}x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \\ x_{31} & x_{32} & x_{33}\end{bmatrix} \mapsto \begin{bmatrix}ax_{11}+bx_{22}+cx_{33} & -x_{12} & -x_{13} \\ -x_{21} & cx_{11}+ax_{22}+bx_{33} & -x_{23} \\ -x_{31} & -x_{32} & bx_{11}+cx_{22}+ax_{33}\end{bmatrix},</math> | ||
| − | where $a,b,c$ are given real numbers. | + | where $a,b,c$ are given real numbers. This map is positive if and only if $a \geq 1$, $a + b + c \geq 3$, and $bc \geq (2-a)^2$ whenever $1 \leq a \leq 2$<ref>S. J. Cho, S.-H. Kye, and S. G. Lee. Generalized Choi maps in three-dimensional matrix algebra. ''Linear Algebra Appl.'', 171:213, 1992.</ref>. |
==Syntax== | ==Syntax== | ||
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-1 0 0 0 -1 0 0 0 1 | -1 0 0 0 -1 0 0 0 1 | ||
</pre> | </pre> | ||
| + | |||
| + | ==References== | ||
| + | <references /> | ||
Revision as of 17:33, 4 March 2014
| ChoiMap | |
| Produces the Choi map or one of its generalizations | |
| Other toolboxes required | iden MaxEntangled opt_args |
|---|---|
ChoiMap is a function that returns the Choi matrix of the linear map on $3 \times 3$ matrices that acts as follows:
\[\begin{bmatrix}x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \\ x_{31} & x_{32} & x_{33}\end{bmatrix} \mapsto \begin{bmatrix}ax_{11}+bx_{22}+cx_{33} & -x_{12} & -x_{13} \\ -x_{21} & cx_{11}+ax_{22}+bx_{33} & -x_{23} \\ -x_{31} & -x_{32} & bx_{11}+cx_{22}+ax_{33}\end{bmatrix},\]
where $a,b,c$ are given real numbers. This map is positive if and only if $a \geq 1$, $a + b + c \geq 3$, and $bc \geq (2-a)^2$ whenever $1 \leq a \leq 2$[1].
Syntax
- C = ChoiMap()
- C = ChoiMap(A,B,C)
Argument descriptions
- A,B,C: Real parameters of the Choi map. If they are not provided, the default Choi map (with A = B = 1 and C = 0) is returned.
Examples
The standard Choi map
The following code returns the Choi matrix of the Choi map:
>> ChoiMap()
ans =
1 0 0 0 -1 0 0 0 -1
0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0
-1 0 0 0 1 0 0 0 -1
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0
-1 0 0 0 -1 0 0 0 1
References
- ↑ S. J. Cho, S.-H. Kye, and S. G. Lee. Generalized Choi maps in three-dimensional matrix algebra. Linear Algebra Appl., 171:213, 1992.