CBNorm: Difference between revisions
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{{Function | {{Function | ||
|name=CBNorm | |name=CBNorm | ||
|desc=Computes the | |desc=Computes the completely bounded norm of a superoperator | ||
|req=[http://cvxr.com/cvx/ cvx] | |req=[http://cvxr.com/cvx/ cvx] | ||
|rel=[[DiamondNorm]] | |rel=[[DiamondNorm]] | ||
|cat=[[List of functions#Norms|Norms]] | |cat=[[List of functions#Norms|Norms]] | ||
|upd= | |upd=November 21, 2014}} | ||
<tt>'''CBNorm'''</tt> is a [[List of functions|function]] that computes the completely bounded (CB) norm $\|\Phi\|_{cb}$ of a superoperator $\Phi$. | |||
<tt>'''CBNorm'''</tt> is a [[List of functions|function]] that computes the | |||
==Syntax== | ==Syntax== | ||
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==Argument descriptions== | ==Argument descriptions== | ||
* <tt>PHI</tt>: A superoperator. Should be provided as either a | * <tt>PHI</tt>: 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). | ||
* <tt>DIM</tt> (optional, default has input and output spaces of equal dimension): A 1-by-2 vector containing the input and output dimensions of <tt>PHI</tt>, in that order (equivalently, these are the dimensions of the first and second subsystems of the Choi matrix <tt>PHI</tt>, in that order). If the input or output space is not square, then <tt>DIM</tt>'s first row should contain the input and output row dimensions, and its second row should contain its input and output column dimensions. <tt>DIM</tt> is required if and only if <tt>PHI</tt> has unequal input and output dimensions and is provided as a Choi matrix. | * <tt>DIM</tt> (optional, default has input and output spaces of equal dimension): A 1-by-2 vector containing the input and output dimensions of <tt>PHI</tt>, in that order (equivalently, these are the dimensions of the first and second subsystems of the Choi matrix <tt>PHI</tt>, in that order). If the input or output space is not square, then <tt>DIM</tt>'s first row should contain the input and output row dimensions, and its second row should contain its input and output column dimensions. <tt>DIM</tt> is required if and only if <tt>PHI</tt> has unequal input and output dimensions and is provided as a Choi matrix. | ||
==Examples== | ==Examples== | ||
===Relationship with the | ===Relationship with the diamond norm=== | ||
The CB norm of a superoperator $\Phi$ is equal to the | The CB norm of a superoperator $\Phi$ is equal to the diamond norm of the dual map $\Phi^\dagger$: | ||
<syntaxhighlight> | <syntaxhighlight> | ||
>> 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]}; | >> 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]}; | ||
| Line 34: | Line 33: | ||
19.5928 | 19.5928 | ||
</syntaxhighlight> | </syntaxhighlight> | ||
===Can be used in CVX=== | |||
Just like the <tt>[[DiamondNorm]]</tt> function, <tt>CBNorm</tt> is a convex function that can be used within CVX optimization problems. See the [[DiamondNorm#Can_be_used_within_CVX|example on the <tt>DiamondNorm</tt> documentation page]]. | |||
{{SourceCode|name=CBNorm}} | {{SourceCode|name=CBNorm}} | ||
Revision as of 21:35, 27 November 2014
| CBNorm | |
| Computes the completely bounded norm of a superoperator | |
| Other toolboxes required | cvx |
|---|---|
| Related functions | DiamondNorm |
| Function category | Norms |
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 =
19.5928Can 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.