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Computes the completely bounded norm of a superoperator

Other toolboxes required CVX
Related functions DiamondNorm
Function category Norms
Usable within CVX? yes (convex)

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


  • 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.


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 =


>> DiamondNorm(DualMap(Phi))

ans =


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