# DephasingChannel

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 Other toolboxes required DephasingChannel Produces a dephasing channel none DepolarizingChannel Superoperators

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

$\Delta(X) := (1-p)\mathrm{diag}(X) + pX,$

where $\mathrm{diag}$ is the map that erases everything off the diagonal of its input, and $0 \leq p \leq 1$ is a given parameter ($p = 0$ by default).

## Syntax

• DELTA = DephasingChannel(DIM)
• DELTA = DephasingChannel(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 dephasing channel to produce. P = 0 gives the completely dephasing channel, and P = 1 gives the identity channel.

## Examples

### The completely dephasing channel

The completely dephasing channel maps every density matrix to its diagonal:

>> rho = RandomDensityMatrix(3)

rho =

0.2575            -0.1464 + 0.0460i   0.0869 - 0.1260i
-0.1464 - 0.0460i   0.5016            -0.0074 + 0.1864i
0.0869 + 0.1260i  -0.0074 - 0.1864i   0.2409

>> ApplyMap(rho,DephasingChannel(3))

ans =

0.2575         0         0
0    0.5016         0
0         0    0.2409

## Source code

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

1. %%  DEPHASINGCHANNEL    Produces a dephasing channel
2. %   This function has one required argument:
3. %     DIM: the dimensionality on which the channel acts
4. %
5. %   DELTA = DephasingChannel(DIM) is the Choi matrix of the completely
6. %   dephasing channel that acts on DIM-by-DIM matrices.
7. %
8. %   This function has one optional argument:
9. %     P (default 0)
10. %
11. %   DELTA = DephasingChannel(DIM,P) produces the partially dephasing
12. %   channel (1-P)*D + P*ID, where D is the completely dephasing channel
13. %   and ID is the identity channel.
14. %
15. %   URL: http://www.qetlab.com/DephasingChannel
16.
17. %   requires: iden.m, MaxEntangled.m, opt_args.m
18. %   author: Nathaniel Johnston (nathaniel@njohnston.ca)
19. %   last updated: January 22, 2015
20.
21. function delta = DephasingChannel(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)*diag(diag(psi*psi')) + p*(psi*psi');