Difference between revisions of "ChoiMap"
| Line 20: | Line 20: | ||
==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]]): |
| − | <pre> | + | <pre<noinclude></noinclude>> |
| − | >> 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 35: | ||
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|IsBlockPositive(C)]] % verify that the Choi map is positive | ||
| + | |||
| + | ans = | ||
| + | |||
| + | 1 | ||
| + | </pre<noinclude></noinclude>> | ||
| + | |||
| + | ===The 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 $a = 0$, $b = c = 1$: | ||
| + | <pre> | ||
| + | >> 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 | ||
</pre> | </pre> | ||
==References== | ==References== | ||
<references /> | <references /> | ||
Revision as of 17:44, 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 $a \geq 0$, $a + b + c \geq 2$, 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 =
1
The 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
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.