Code accompanying the manuscript Quantum 5, 522 (2021) (arXiv:2102.07773).
The MATLAB code below shows an implementation of the SDPs for the various measures described in the manuscript. It requires CVX to run the optimisation problems and uses the helper function PartialTrace from QETLAB. The programs all have the same structure: as an argument, they take the Choi matrix J of a given Hermiticity-preserving map and optionally a tuple of input/output dimensions [d_A, d_B] (assumed to be equal if not provided). In addition to the optimal value, they return an optimal decomposition of J (depending on the definition of the given measure) and an optimal dual witness W which can be used to certify the optimal value (as per the dual forms of the programs given in the manuscript).
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function [cvx_optval,X,Y,W] = Diamond(J,dim)
J = (J+J')/2; % make sure that J is Hermitian, avoid numerical issues
if nargin<2
d = max(size(J));
dim = [1,1]*sqrt(d);
else
d = dim(1)*dim(2);
end
cvx_begin sdp quiet
variable X(d,d) hermitian semidefinite
variable Y(d,d) hermitian semidefinite
variable mu nonnegative
dual variable W
PartialTrace(X+Y,2,dim) <= mu*eye(dim(1));
W: X-Y == J;
minimize mu
cvx_end
end
function [cvx_optval,X,Y,W] = NormCPTNI(J,dim)
J = (J+J')/2; % make sure that J is Hermitian, avoid numerical issues
if nargin<2
d = max(size(J));
dim = [1,1]*sqrt(d);
else
d = dim(1)*dim(2);
end
cvx_begin sdp quiet
variable X(d,d) hermitian semidefinite
variable Y(d,d) hermitian semidefinite
variable mu1 nonnegative
variable mu2 nonnegative
dual variable W
PartialTrace(X,2,dim) <= mu1*eye(dim(1));
PartialTrace(Y,2,dim) <= mu2*eye(dim(1));
W: X-Y == J;
minimize mu1+mu2
cvx_end
end
function [cvx_optval,X,W,Y] = RobCPTNI(J,dim)
J = (J+J')/2; % make sure that J is Hermitian, avoid numerical issues
if nargin<2
d = max(size(J));
dim = [1,1]*sqrt(d);
else
d = dim(1)*dim(2);
end
cvx_begin sdp quiet
variable X(d,d) hermitian semidefinite
variable lam nonnegative
dual variable Z
dual variable Y
PartialTrace(X,2,dim) <= lam*eye(dim(1));
Y: PartialTrace(J+X,2,dim) <= (1+lam)*eye(dim(1));
Z: J + X >=0
minimize lam
cvx_end
W = kron(Y,eye(dim(2)))-Z; % notation from the paper
end
function [cvx_optval,X,W] = RobCPTNI2(J,dim)
J = (J+J')/2; % make sure that J is Hermitian, avoid numerical issues
if nargin<2
d = max(size(J));
dim = [1,1]*sqrt(d);
else
d = dim(1)*dim(2);
end
cvx_begin sdp quiet
variable X(d,d) hermitian semidefinite
variable lam nonnegative
dual variable W
PartialTrace(X,2,dim) <= (1+lam)*eye(dim(1));
W : X >= J
minimize lam
cvx_end
end