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I am trying to solve a system of four matrix equations using Reduce function. The system arises in Quantum Information. The system is solvable with a solution i.e. phi, but the following code gives false to the presence of solution. Can you suggest where am I going wrong and what is to be done to fix this?

Following is the full code I am using:

   H = {{1, 1}, {1, -1}};
H2 = {{1, 1}, {I, -I}};
II = {{1, 0}, {0, 1}};
B1 = KroneckerProduct[II, H];
B2 = KroneckerProduct[II, H2];
B3 = KroneckerProduct[H,II];
B4 = KroneckerProduct[H2,II];
{B1 // MatrixForm, B2 // MatrixForm, B3 // MatrixForm, B4 // MatrixForm}

randomPureState[n_] := RandomComplex[{-1 - I, 1 + I}, n] // Normalize
singleStateDensityMatrix[state_] := Outer[Times, state, Conjugate[state]]

randomPureDensityMatrix[n_] := singleStateDensityMatrix@randomPureState@n

phi = randomPureState[4]
X = singleStateDensityMatrix[phi]

Pr1 = ((ConjugateTranspose[B1]) . X . B1);
{Pr1 // MatrixForm}
Pr2 = ((ConjugateTranspose[B2]) . X . B2);
{Pr2 // MatrixForm}
Pr3 = ((ConjugateTranspose[B3]) . X . B3);
{Pr3 // MatrixForm}
Pr4 = ((ConjugateTranspose[B4]) . X . B4);
{Pr4 // MatrixForm}

diagonalPrB1 = Diagonal[Pr1]
diagonalPrB2 = Diagonal[Pr2]
diagonalPrB3 = Diagonal[Pr3]
diagonalPrB4 = Diagonal[Pr4]



psi = Array[y, {4, 1}];
Reduce[{Abs[(ConjugateTranspose[B1] . psi)]^2 == diagonalPrB1, 
  Abs[(ConjugateTranspose[B2] . psi)]^2 == diagonalPrB2,
  Abs[(ConjugateTranspose[B3] . psi)]^2 == diagonalPrB3, 
  Abs[(ConjugateTranspose[B4] . psi)]^2 == diagonalPrB4}, Flatten[psi]]
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    $\begingroup$ Are you sure that the system is solvable with any random input though? Have you tried temporarily replacing your random-based entries with known good values and seeing if a solution can be found in that case? $\endgroup$ Commented Jun 26, 2023 at 9:13
  • $\begingroup$ B3,B4 =B1,B1? What is y? Perhaps its better not to suppress definition echos by ; $\endgroup$ Commented Jun 26, 2023 at 12:39
  • $\begingroup$ Sorry for my mistake. B1 and B2 are not the same as B3 and B4. I have edited them in the question. And yes, the system is solvable for all the phi. $\endgroup$ Commented Jun 26, 2023 at 14:41
  • $\begingroup$ Once again, have you tried replacing the random numbers with something simple and trying to solve your equations by hand, just to check that they are correct? Or even better, try plugging in a known correct solution to your equations, to check that they are correct. If you don't do your own basic troubleshooting, it's unlikely that you will find somebody to do it for you with little context of the problem. $\endgroup$ Commented Jun 26, 2023 at 15:50
  • $\begingroup$ In the code my objective is to take probabilities of phi in the bases B1, B2, B3 and B4. Using these probabilities, I am trying to reconstruct the original state phi in the name of psi. I know with certainty that the equations have a solution for any phi as we have mathematical proof of this. I want to know, why does Reduce show that there is no solution.? Am I using the function in the wrong way or there is some other issue? I have tried to use the reconstruction for specific column matrices as well, but the problem is still there. $\endgroup$ Commented Jun 27, 2023 at 6:01

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