Superoperator dims

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superoperator_dims
Computes the input, output, and environment dimensions of a superoperator

Other toolboxes required none
Function category 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 on "expand" to the right to view the MATLAB source code for this function.