# ApplyMap

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 Other toolboxes required ApplyMap Applies a superoperator to an operator none PartialMap

ApplyMap is a function that applies a superoperator to an operator. Both the superoperator and the operator may be either full or sparse.

## Syntax

• PHIX = ApplyMap(X,PHI)

## Argument descriptions

• X: A matrix.
• 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).

## Examples

### A random example

The following code computes $\Phi(X)$, where $X = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}$ and $\Phi$ is the superoperator defined by $$\Phi(X) = \begin{bmatrix}1 & 5 \\ 1 & 0 \\ 0 & 2\end{bmatrix}X\begin{bmatrix}0 & 1 \\ 2 & 3 \\ 4 & 5\end{bmatrix}^\dagger - \begin{bmatrix}1 & 0 \\ 0 & 0 \\ 0 & 1\end{bmatrix}X\begin{bmatrix}0 & 0 \\ 1 & 1 \\ 0 & 0\end{bmatrix}^\dagger.$$

>> X = [1 2;3 4];
>> Phi = {[1 5;1 0;0 2] [0 1;2 3;4 5];[-1 0;0 0;0 -1] [0 0;1 1;0 0]};
>> ApplyMap(X,Phi)

ans =

22    95   174
2     8    14
8    29    64


### Transpose map

The swap operator is the Choi matrix of the transpose map. Thus, the following code is a (rather slow and ugly) way of computing the transpose of a matrix:

>> X = reshape(1:9,3,3)

X =

1     4     7
2     5     8
3     6     9

>> ApplyMap(X,SwapOperator(3))

ans =

1     2     3
4     5     6
7     8     9


Of course, in practice you should just use MATLAB's built-in transposition operator X.'.