# Superoperator dims

 Other toolboxes required superoperator_dims Computes the input, output, and environment dimensions of a superoperator none Helper functions
 This is a helper function that only exists to aid other functions in QETLAB. If you are an end-user of QETLAB, you likely will never have a reason to use this function.

superoperator_dims is a 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

• [DA,DB,DE] = superoperator_dims(PHI)
• [DA,DB,DE] = superoperator_dims(PHI,ALLOW_RECT)
• [DA,DB,DE] = superoperator_dims(PHI,ALLOW_RECT,DIM)

## Argument descriptions

### Input arguments

• PHI: A superoperator, represented either as a Choi matrix or as a cell of Kraus operators.
• ALLOW_RECT (optional, default 1): A flag (either 1 or 0) indicating that the input and output spaces of PHI can or can't be rectangular (non-square). If ALLOW_RECT == 0 and PHI has non-square input or output space, an error is produced.
• DIM (optional): A 1-by-2 vector containing the dimensions of the input and output space of PHI. 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

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

## 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):

```>> Phi = KrausOperators(RandomSuperoperator([3,4]),[3,4]);
>> [da,db,de] = superoperator_dims(Phi)

da =

3     3

db =

4     4

de =

12```

## Source code

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