# CBNorm

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CBNorm | |

Computes the completely bounded norm of a superoperator | |

Other toolboxes required | CVX |
---|---|

Related functions | DiamondNorm InducedSchattenNorm |

Function category | Norms |

Usable within CVX? | yes (convex) |

` CBNorm` is a function that computes the completely bounded (CB) norm $\|\Phi\|_{cb}$ of a superoperator $\Phi$.

## Syntax

`CB = CBNorm(PHI)``CB = CBNorm(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).`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

### Relationship with the diamond norm

The CB norm of a superoperator $\Phi$ is equal to the diamond norm of the dual map $\Phi^\dagger$:

```
>> Phi = {[1 2;3 4],[0 1;1 0] ; [0 1;2 0],[1 1;1 1] ; [1 1;-1 3],[1 4;0 0]};
>> CBNorm(Phi)
ans =
19.5928
>> DiamondNorm(DualMap(Phi))
ans =
19.5928
```

### Can be used in CVX

Just like the `DiamondNorm` function, `CBNorm` is a convex function that can be used within CVX optimization problems. See the example on the `DiamondNorm` documentation page.

## Source code

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