DepolarizingChannel

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DepolarizingChannel
Produces a depolarizing channel

Other toolboxes required none
Related functions DephasingChannel
Function category Superoperators

DepolarizingChannel is a function that returns the Choi matrix of the partially depolarizing channel, which acts as follows:

\[\Delta(X) := (1-p)\mathrm{Tr}(X)\frac{I}{d^2} + pX,\]

where $I$ is the identity matrix, $d$ is the local dimension, and $0 \leq p \leq 1$ is a given parameter ($p = 0$ by default).

Syntax

  • DELTA = DepolarizingChannel(DIM)
  • DELTA = DepolarizingChannel(DIM,P)

Argument descriptions

  • DIM: The dimension of the channel. That is, the channel will act on DIM-by-DIM matrices.
  • P (optional, default 0): A parameter (from 0 to 1, inclusive) that specifies which partially depolarizing channel to produce. P = 0 gives the completely depolarizing channel, and P = 1 gives the identity channel.

Examples

The completely depolarizing channel

The completely depolarizing channel maps every density matrix to the maximally-mixed state:

>> ApplyMap(RandomDensityMatrix(3),DepolarizingChannel(3))
 
ans =
 
    0.3333         0         0
         0    0.3333         0
         0         0    0.3333

Source code

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

  1. %%  DEPOLARIZINGCHANNEL    Produces a depolarizing channel
  2. %   This function has one required argument:
  3. %     DIM: the dimensionality on which the channel acts
  4. %
  5. %   DELTA = DepolarizingChannel(DIM) is the Choi matrix of the completely
  6. %   depolarizing channel that acts on DIM-by-DIM matrices.
  7. %
  8. %   This function has one optional argument:
  9. %     P (default 0)
  10. %   
  11. %   DELTA = DepolarizingChannel(DIM,P) produces the partially depolarizing
  12. %   channel (1-P)*D + P*ID, where D is the completely depolarizing channel
  13. %   and ID is the identity channel.
  14. %
  15. %   URL: http://www.qetlab.com/DepolarizingChannel
  16.  
  17. %   requires: iden.m, MaxEntangled.m, opt_args.m
  18. %   author: Nathaniel Johnston (nathaniel@njohnston.ca)
  19. %   last updated: March 4, 2014
  20.  
  21. function delta = DepolarizingChannel(dim,varargin)
  22.  
  23. % set optional argument defaults: p=0
  24. [p] = opt_args({ 0 },varargin{:});
  25.  
  26. % compute the Choi matrix of the depolarizing channel
  27. psi = MaxEntangled(dim,1,0); % gives a sparse non-normalized state
  28. delta = (1-p)*speye(dim^2)/dim + p*(psi*psi');