KrausOperators.m

%%  KRAUSOPERATORS    Computes a set of Kraus operators for a superoperator
%   This function has one required input argument:
%     PHI: a superoperator
%
%   KO = KrausOperators(PHI) is a cell containing Kraus operators for PHI.
%   This set of Kraus operators will always be canonical in the following
%   ways:
%     (1) If PHI is completely positive, only the left Kraus operators will
%     be returned (and the right Kraus operators are the same).
%     (2) If PHI is Hermiticity preserving, the right Kraus operators are
%     the same as the left Kraus operators, up to sign. The pairs of Kraus
%     operators that are equal are given first, followed by the pairs that
%     are negatives of each other.
%     (3) The left Kraus operators form an orthogonal set in the
%     Hilbert-Schmidt inner product, and similarly for the right Kraus
%     operators.
%
%   This function has one optional input argument:
%     DIM (default has both subsystems of equal dimension)
%
%   KO = KrausOperators(PHI,DIM) is the same as above, where DIM is a
%   1-by-2 vector containing the input and output dimensions of PHI, in
%   that order (equivalently, these are the dimensions of the first and
%   second subsystems of the Choi matrix PHI, in that order). DIM is
%   required if and only if PHI has unequal input and output dimensions and
%   is provided as a Choi matrix.
%
%   URL: http://www.qetlab.com/KrausOperators

%   requires: ApplyMap.m, ChoiMatrix.m, iden.m, IsCP.m, IsHermPreserving.m,
%             IsPSD.m, MaxEntangled.m, opt_args.m, PermuteSystems.m,
%             sporth.m, superoperator_dims.m
%
%   author: Nathaniel Johnston (nathaniel@njohnston.ca)
%   package: QETLAB
%   last updated: July 16, 2018

function ko = KrausOperators(Phi,varargin)

% Compute the dimensions of PHI.
[da,db] = superoperator_dims(Phi,1,varargin{:});
max_dim = max(da(1)*db(1),da(2)*db(2));

% If PHI is already a set of Kraus operators, still do some work: we want a
% *canonical* set of Kraus operators, so convert everything to a Choi
% matrix first.
Phi = ChoiMatrix(Phi); % if PHI is a Choi matrix already, this does no work: don't worry
if(norm(Phi,'fro') < da(1)^2*db(1)^2*eps)
    ko = {zeros(db(1),da(1)),zeros(db(2),da(2))}; % need to handle this case separately, or no Kraus operators are returned
    return
end

% Compute a canonical set of Kraus operators for Hermiticity-preserving
% maps.
if(IsHermPreserving(Phi))
    ep = max_dim*eps(norm(Phi,'fro'));
    [V,S] = eig(full(Phi));
    [S,ind] = sort(diag(S),'descend');
    V = V(:,ind);
    
    ind1 = find(S <= ep,1) - 1;
    ind2 = find(S < -ep,1) - 1;
    if(min(size(ind1)) == 0)
        ind1 = length(S);
    end
    if(min(size(ind2)) == 0)
        ind2 = length(S);
    end
    
    sgnS = sign(S);
    V = V*diag(sqrt(abs(S)));
    ko(:,1) = mat2cell(reshape(V(:,[1:ind1,ind2+1:end]),db(1),da(1)*(ind1+length(S)-ind2)),db(1),da(1)*ones((ind1+length(S)-ind2),1)).';
    if(~IsCP(Phi))
        V = V*diag(sgnS);
        ko(:,2) = mat2cell(reshape(V(:,[1:ind1,ind2+1:end]),db(1),da(1)*(ind1+length(S)-ind2)),db(1),da(1)*ones((ind1+length(S)-ind2),1)).';
    end
    
% Compute a canonical set of Kraus operators for all other maps.
else
    [U,S,V] = svd(full(Phi));
    S = diag(S);
    ind = find(S <= max_dim*eps(norm(Phi,'fro')),1) - 1;
    if(min(size(ind)) == 0)
        ind = length(S);
    end
    S = diag(sqrt(S));
    U = U*pad_array(S,size(U,2)-size(S,1),1);
    V = V*pad_array(S,size(V,2)-size(S,1),1);

    ko(:,1) = mat2cell(reshape(U(:,1:ind),db(1),da(1)*ind),db(1),da(1)*ones(ind,1)).';
    ko(:,2) = mat2cell(reshape(V(:,1:ind),db(2),da(2)*ind),db(2),da(2)*ones(ind,1)).';
end