ReductionMap
From QETLAB
ReductionMap | |
Produces the reduction map | |
Other toolboxes required | none |
---|---|
Related functions | ChoiMap |
Function category | Superoperators |
ReductionMap is a function that returns the Choi matrix of the linear map that acts as follows:
\[\Phi(X) := \mathrm{Tr}(X)I - X,\]
where $I$ is the identity matrix. This map is positive.
Contents
Syntax
- R = ReductionMap(DIM)
- R = ReductionMap(DIM,K)
Argument descriptions
- DIM: The dimension of the reduction map. That is, the size of the matrices that the reduction map acts on.
- K (optional, default 1): If this positive integer is provided, the script will instead return the Choi matrix of the following linear map:
\[\Phi(X) := K\cdot\mathrm{Tr}(X)I - X.\]
Examples
The reduction map is positive
The following code returns the Choi matrix of the 3-dimensional reduction map and then verifies that the reduction map is indeed positive (i.e., verifies that its Choi matrix is block positive):
>> R = ReductionMap(3) R = (5,1) -1 (9,1) -1 (2,2) 1 (3,3) 1 (4,4) 1 (1,5) -1 (9,5) -1 (6,6) 1 (7,7) 1 (8,8) 1 (1,9) -1 (5,9) -1 >> IsBlockPositive(R) % verify that the reduction map is positive ans = 1
Higher values of K
It is known that the generalization of the reduction map that is provided by the optional argument K is always K-positive, but not (K+1)-positive. The following code verifies this in the K = 2 case:
>> R = ReductionMap(3,2) R = (1,1) 1 (5,1) -1 (9,1) -1 (2,2) 2 (3,3) 2 (4,4) 2 (1,5) -1 (5,5) 1 (9,5) -1 (6,6) 2 (7,7) 2 (8,8) 2 (1,9) -1 (5,9) -1 (9,9) 1 >> IsBlockPositive(R,1) % verify that this map is positive ans = 1 >> IsBlockPositive(R,2) % verify that this map is 2-positive ans = 1 >> IsBlockPositive(R,3) % see that this map is not 3-positive ans = 0
Source code
Click on "expand" to the right to view the MATLAB source code for this function.
%% REDUCTIONMAP Produces the reduction map
% This function has one required argument:
% DIM: a positive integer (the dimension of the reduction map)
%
% R = ReductionMap(DIM) is the Choi matrix of the reduction map, which is
% a positive map on DIM-by-DIM matrices.
%
% This function has one optional argument:
% K (default 1)
%
% R = ReductionMap(DIM,K) is the Choi matrix of the map defined by
% R(X) = K*trace(X)*eye(DIM^2) - X. This map is K-positive.
%
% URL: http://www.qetlab.com/ReductionMap
% requires: iden.m, MaxEntangled.m, opt_args.m
% author: Nathaniel Johnston (nathaniel@njohnston.ca)
% package: QETLAB
% last updated: September 29, 2014
function R = ReductionMap(dim,varargin)
% set optional argument defaults: k=1 (the usual reduction map)
[k] = opt_args({ 1 },varargin{:});
psi = MaxEntangled(dim,1,0);
R = k*speye(dim^2) - psi*psi';