# ReductionMap

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### Higher values of

ReductionMap | |

Produces the reduction map | |

Other toolboxes required | none |
---|---|

Related functions | ChoiMap |

Function category | Superoperators |

` ReductionMap` is a function that returns the Choi matrix of the linear map that acts as follows:

\[\Phi(X) := \mathrm{Tr}(X)I - X,\]

where $I$ is the identity matrix. This map is positive.

## Syntax

`R = ReductionMap(DIM)``R = ReductionMap(DIM,K)`

## Argument descriptions

`DIM`: The dimension of the reduction map. That is, the size of the matrices that the reduction map acts on.`K`(optional, default 1): If this positive integer is provided, the script will instead return the Choi matrix of the following linear map:

\[\Phi(X) := K\cdot\mathrm{Tr}(X)I - X.\]

## Examples

### The reduction map is positive

The following code returns the Choi matrix of the 3-dimensional reduction map and then verifies that the reduction map is indeed positive (i.e., verifies that its Choi matrix is block positive):

```
>> R = ReductionMap(3)
R =
(5,1) -1
(9,1) -1
(2,2) 1
(3,3) 1
(4,4) 1
(1,5) -1
(9,5) -1
(6,6) 1
(7,7) 1
(8,8) 1
(1,9) -1
(5,9) -1
>> IsBlockPositive(R) % verify that the reduction map is positive
ans =
1
```

### Higher values of `K`

It is known that the generalization of the reduction map that is provided by the optional argument `K` is always `K`-positive, but not (`K+1`)-positive. The following code verifies this in the `K = 2` case:

```
>> R = ReductionMap(3,2)
R =
(1,1) 1
(5,1) -1
(9,1) -1
(2,2) 2
(3,3) 2
(4,4) 2
(1,5) -1
(5,5) 1
(9,5) -1
(6,6) 2
(7,7) 2
(8,8) 2
(1,9) -1
(5,9) -1
(9,9) 1
>> IsBlockPositive(R,1) % verify that this map is positive
ans =
1
>> IsBlockPositive(R,2) % verify that this map is 2-positive
ans =
1
>> IsBlockPositive(R,3) % see that this map is not 3-positive
ans =
0
```

## Source code

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