## Finding examples of sets of POVMs that require a maximal parent

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In [2]:

alpha = [1,sqrt(2)/4];
beta = [0,sqrt(2)/4];
gam = [0,sqrt(3)/2];

rhoxlam = zeros(3,3,3,8);
omegalam = zeros(3,3,8);

Id = eye(3);

for a0 = 0:1
for a1 = 0:1
for a2 = 0:1
av = [a0 a1 a2];
psi = alpha(a2+1)*(-1)^(a0+a1)*Id(:,a1+1) + beta(a2+1)*Id(:,~a1+1) + gam(a2+1)*(-1)^(a0*a1)*Id(:,3);
omegalam(:,:,1+a2+2*a1+4*a0) = (~a2*(~a0+~a1)+a2*(~a0+~a1-1))/5*psi*psi';
for x = 0:2
rhoxlam(:,:,x+1,1+a2+2*a1+4*a0) = -1/5*(-1)^av(x+1)*psi*psi';
end
end
end
end

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In [27]:

theta = pi/8;
phi = asin(sqrt(2/3)); % needs to be this for a valid POVM

ct = cos(theta);
st = sin(theta);

cp = cos(phi);
sp = sin(phi);

X = Pauli(1);
Z = Pauli(3);

Ga = zeros(3,3,8);

Ga(1:2,1:2,1) = 3/8*[ct;st]*[ct,st];
Ga(1:2,1:2,3) = 3/8*[st;ct]*[st,ct];
Ga(1:2,1:2,5) = 3/8*[-ct;st]*[-ct,st];
Ga(1:2,1:2,7) = 3/8*[-st;ct]*[-st,ct];

Ga(:,:,2) = 3/8*[cp/sqrt(2);cp/sqrt(2);sp]*[cp/sqrt(2),cp/sqrt(2),sp];
Ga(:,:,4) = 3/8*[cp/sqrt(2);-cp/sqrt(2);sp]*[cp/sqrt(2),-cp/sqrt(2),sp];
Ga(:,:,6) = 3/8*[cp/sqrt(2);-cp/sqrt(2);-sp]*[cp/sqrt(2),-cp/sqrt(2),-sp];
Ga(:,:,8) = 3/8*[cp/sqrt(2);cp/sqrt(2);-sp]*[cp/sqrt(2),cp/sqrt(2),-sp];

sum(Ga,3);

Max = zeros(3,3,3);

Max(:,:,1) = sum(Ga(:,:,[1,2,3,4]),3);
Max(:,:,2) = sum(Ga(:,:,[1,2,5,6]),3);
Max(:,:,3) = sum(Ga(:,:,[1,3,5,7]),3);

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In [28]:

for i = 1:8
rho = rhoxlam(:,:,:,i);
omega = omegalam(:,:,i);
mu(i) = real(sumall(rho.*conj(Max))+trace(omega));
end

mu

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mu =

Columns 1 through 7

-0.0030   -0.0370   -0.0030   -0.0105   -0.0030   -0.0370   -0.0030

Column 8

-0.0105

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In [ ]:

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