# ComplementaryMap

ComplementaryMap | |

Computes the complementary map of a superoperator | |

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

Related functions | DualMap |

Function category | Superoperators |

` ComplementaryMap` is a function that computes the complementary map of a superoperator (in the sense that the output of this function describes the information leaked by the original superoperator to the environment).

## Syntax

`PHIC = ComplementaryMap(PHI)``PHIC = ComplementaryMap(PHI,DIM)`

## Argument descriptions

`PHI`: A superoperator. Should be provided as either a Choi matrix, or as a cell with either 1 or 2 columns (see the tutorial page for more details about specifying superoperators within QETLAB).`PHIC`will be a cell of Kraus operators if`PHI`is a cell of Kraus operators, and similarly`PHIC`will be a Choi matrix if`PHI`is a Choi matrix.`DIM`(optional, default has input and output spaces of equal dimension): A 1-by-2 vector containing the input and output dimensions of`PHI`, in that order (equivalently, these are the dimensions of the first and second subsystems of the Choi matrix`PHI`, in that order). If the input or output space is not square, then`DIM`'s first row should contain the input and output row dimensions, and its second row should contain its input and output column dimensions.`DIM`is required if and only if`PHI`has unequal input and output dimensions and is provided as a Choi matrix.

## Examples

### Non-uniqueness

Complementary maps are not unique, and hence different maps `PHIC` may be returned depending on the particular representation of the input map `PHI`. The particular complementary map that is returned by this function is the one that is obtained by placing all of the first rows of Kraus operators of `PHI` into the first Kraus operator of `PHIC`, all of the second rows of Kraus operators of `PHI` into the second Kraus operator of `PHIC`, and so on. The following code defines two families of Kraus operators `Phi` and `Phi2`, verifies that they represent the same map by showing that their Choi matrices are the same, and then shows that nonetheless the different Kraus representations lead to different complementary maps.

```
>> Phi = {[1 0;0 0] ; [0 1;0 0] ; [0 0;1 0] ; [0 0;0 1]};
>> Phi2 = {[1 0;0 1]/sqrt(2) ; [0 1;1 0]/sqrt(2) ; [0 -1i;1i 0]/sqrt(2) ; [1 0;0 -1]/sqrt(2)};
>> ChoiMatrix(Phi)
ans =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
>> ChoiMatrix(Phi2)
ans =
1.0000 0 0 0
0 1.0000 0 0
0 0 1.0000 0
0 0 0 1.0000
>> celldisp(ComplementaryMap(Phi))
ans{1} =
1 0
0 1
0 0
0 0
ans{2} =
0 0
0 0
1 0
0 1
>> celldisp(ComplementaryMap(Phi2))
ans{1} =
0.7071 0
0 0.7071
0 0 - 0.7071i
0.7071 0
ans{2} =
0 0.7071
0.7071 0
0 + 0.7071i 0
0 -0.7071
```

## Source code

Click on "expand" to the right to view the MATLAB source code for this function.