Nathaniel: Created page with "{{Function |name=superoperator_dims |desc=Computes the input, output, and environment dimensions of a superoperator |cat=[[List of functions#Helper_functions|Helper functions]..."

2014-11-27T21:10:38Z

Created page with "{{Function |name=superoperator_dims |desc=Computes the input, output, and environment dimensions of a superoperator |cat=[[List of functions#Helper_functions|Helper functions]..."

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{{Function
|name=superoperator_dims
|desc=Computes the input, output, and environment dimensions of a superoperator
|cat=[[List of functions#Helper_functions|Helper functions]]
|upd=November 27, 2014
|helper=1}}
<tt>'''superoperator_dims'''</tt> is a [[List of functions|function]] that computes all of the dimensions of a given superoperator (i.e., the dimensions of its input space, output space, and environment space). It also serves as an error-checking function that will get cranky if the superoperator passed into it does not really represent a superopertor.

==Syntax==
* <tt>[DA,DB,DE] = superoperator_dims(PHI)</tt>
* <tt>[DA,DB,DE] = superoperator_dims(PHI,ALLOW_RECT)</tt>
* <tt>[DA,DB,DE] = superoperator_dims(PHI,ALLOW_RECT,DIM)</tt>

==Argument descriptions==
===Input arguments===
* <tt>PHI</tt>: A superoperator, represented either as a Choi matrix or as a cell of Kraus operators.
* <tt>ALLOW_RECT</tt> (optional, default 1): A flag (either 1 or 0) indicating that the input and output spaces of <tt>PHI</tt> can or can't be rectangular (non-square). If <tt>ALLOW_RECT == 0</tt> and <tt>PHI</tt> has non-square input or output space, an error is produced.
* <tt>DIM</tt> (optional): A 1-by-2 vector containing the dimensions of the input and output space of <tt>PHI</tt>. If this argument is provided, the function serves only as an error-checking function that makes sure that the computed dimensions agree with these given dimensions (and gives an error otherwise).

===Output arguments===
* <tt>DA</tt>: A 1-by-2 vector containing the row and column dimensions of the input space of <tt>PHI</tt> (if <tt>ALLOW_RECT == 0</tt> then <tt>DA</tt> is a scalar, not a vector).
* <tt>DB</tt> (optional): A 1-by-2 vector containing the row and column dimensions of the output space of <tt>PHI</tt> (if <tt>ALLOW_RECT == 0</tt> then <tt>DB</tt> is a scalar, not a vector).
* <tt>DE</tt> (optional): A scalar indicating the environment dimension of <tt>PHI</tt> (this is sometimes call the "Choi rank" of <tt>PHI</tt>: it is the minimal number of Kraus operators in any representation of <tt>PHI</tt>).

==Examples==
The following code generates a random quantum channel from $M_3$ to $M_4$ and then computes its dimensions, and verifies that it cannot be written with fewer than 12 Kraus operators (which happens with probability 1):
<syntaxhighlight>
>> Phi = KrausOperators(RandomSuperoperator([3,4]),[3,4]);
>> [da,db,de] = superoperator_dims(Phi)

da =

3 3

db =

4 4

de =

12
</syntaxhighlight>

{{SourceCode|name=superoperator_dims|helper=1}}

Superoperator dims - Revision history

Nathaniel: Created page with "{{Function |name=superoperator_dims |desc=Computes the input, output, and environment dimensions of a superoperator |cat=[[List of functions#Helper_functions|Helper functions]..."