Difference between revisions of "ChoiMap"
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|name=ChoiMap | |name=ChoiMap | ||
|desc=Produces the Choi map or one of its generalizations | |desc=Produces the Choi map or one of its generalizations | ||
| − | | | + | |rel=[[ReductionMap]] |
| + | |cat=[[List of functions#Superoperators|Superoperators]] | ||
|upd=August 5, 2013 | |upd=August 5, 2013 | ||
| − | |v= | + | |v=0.50}} |
<tt>'''ChoiMap'''</tt> is a [[List of functions|function]] that returns the Choi matrix of the linear map on $3 \times 3$ matrices that acts as follows: | <tt>'''ChoiMap'''</tt> is a [[List of functions|function]] that returns the Choi matrix of the linear map on $3 \times 3$ matrices that acts as follows: | ||
: <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. This map is positive if and only if $a \geq | + | where $a,b,c$ are given real numbers. This map is positive if and only if $a \geq 0$, $a + b + c \geq 2$, and $bc \geq (1-a)^2$ whenever $0 \leq a \leq 1$<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== | ||
| Line 20: | Line 21: | ||
==Examples== | ==Examples== | ||
===The standard Choi map=== | ===The standard Choi map=== | ||
| − | The following code returns the Choi matrix of the Choi map: | + | The following code returns the Choi matrix of the Choi map and then verifies that the Choi map is indeed positive (i.e., verifies that its Choi matrix is [[block positive]]): |
| − | < | + | <syntaxhighlight> |
| − | >> ChoiMap() | + | >> C = ChoiMap() |
| − | + | C = | |
1 0 0 0 -1 0 0 0 -1 | 1 0 0 0 -1 0 0 0 -1 | ||
| Line 35: | Line 36: | ||
0 0 0 0 0 0 0 1 0 | 0 0 0 0 0 0 0 1 0 | ||
-1 0 0 0 -1 0 0 0 1 | -1 0 0 0 -1 0 0 0 1 | ||
| − | </ | + | |
| + | >> IsBlockPositive(C) % verify that the Choi map is positive | ||
| + | |||
| + | ans = | ||
| + | |||
| + | 1 | ||
| + | </syntaxhighlight> | ||
| + | |||
| + | ===The reduction map=== | ||
| + | The [[ReductionMap|reduction map]] is the map $R$ defined by $R(X) = {\rm Tr}(X)I - X$, where $I$ is the identity operator. The reduction map is the Choi map that arises when $a = 0$, $b = c = 1$: | ||
| + | <syntaxhighlight> | ||
| + | >> ChoiMap(0,1,1) | ||
| + | |||
| + | ans = | ||
| + | |||
| + | 0 0 0 0 -1 0 0 0 -1 | ||
| + | 0 1 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 0 0 0 0 -1 | ||
| + | 0 0 0 0 0 1 0 0 0 | ||
| + | 0 0 0 0 0 0 1 0 0 | ||
| + | 0 0 0 0 0 0 0 1 0 | ||
| + | -1 0 0 0 -1 0 0 0 0 | ||
| + | |||
| + | >> full(ReductionMap(3)) | ||
| + | |||
| + | ans = | ||
| + | |||
| + | 0 0 0 0 -1 0 0 0 -1 | ||
| + | 0 1 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 0 0 0 0 -1 | ||
| + | 0 0 0 0 0 1 0 0 0 | ||
| + | 0 0 0 0 0 0 1 0 0 | ||
| + | 0 0 0 0 0 0 0 1 0 | ||
| + | -1 0 0 0 -1 0 0 0 0 | ||
| + | </syntaxhighlight> | ||
| + | |||
| + | {{SourceCode|name=ChoiMap}} | ||
==References== | ==References== | ||
<references /> | <references /> | ||
Latest revision as of 17:22, 29 September 2014
| ChoiMap | |
| Produces the Choi map or one of its generalizations | |
| Other toolboxes required | none |
|---|---|
| Related functions | ReductionMap |
| Function category | Superoperators |
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 $a \geq 0$, $a + b + c \geq 2$, and $bc \geq (1-a)^2$ whenever $0 \leq a \leq 1$[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 and then verifies that the Choi map is indeed positive (i.e., verifies that its Choi matrix is block positive):
>> C = ChoiMap()
C =
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
>> IsBlockPositive(C) % verify that the Choi map is positive
ans =
1The reduction map
The reduction map is the map $R$ defined by $R(X) = {\rm Tr}(X)I - X$, where $I$ is the identity operator. The reduction map is the Choi map that arises when $a = 0$, $b = c = 1$:
>> ChoiMap(0,1,1)
ans =
0 0 0 0 -1 0 0 0 -1
0 1 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 0 0 0 0 -1
0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0
-1 0 0 0 -1 0 0 0 0
>> full(ReductionMap(3))
ans =
0 0 0 0 -1 0 0 0 -1
0 1 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 0 0 0 0 -1
0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 1 0
-1 0 0 0 -1 0 0 0 0Source code
Click here to view this function's source code on github.
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.