Equation (7) in the 2012 document, "Complementarity reveals the entanglement of two twisted photons" by B. C. Hiesmayr and W. Löffler for a state $ rho_d $ In the "simplex magic" of the bell states.

begin {equation}

rho_d = frac {q_4 (1- delta (d-3)) sum_ {z = 2} ^ {d-2} left ( sum_ {i = 0} ^ {d-1}

P_ {i, z} right)} {d} + frac {q_2 sum _ {i = 1} ^ {d-1} P_ {i, 0}} {(d-1) (d + 1) } + frac {q_3 sum

_ {i = 0} ^ {d-1} P_ {i, 1}} {d} + frac { left (- frac {q_1} {d ^ 2-d-1} – frac {q_2} {d + 1} – (d-3)

q_4-q_3 + 1 right) text {IdentityMatrix} left[d^2right]} {d ^ 2} + frac {q_1

P_ {0,0}} {d ^ 2-d-1}

end {equation}

yields for certain values of the $ q_i $& # 39; s, to $ d = 3 $ the Horodecki state of a single parameter, the first delimited entangled state found ".

More generally, for that matter $ d = 3 $, the restriction that requires partial transposition (obtained by transposing the nine $ 3 times 3 $ blocks) of the density matrix $ rho_3 $ being positive defined takes the form

```
restriction3 = q1> 0 && q2> 0 && q3> 0 && 4 q1 + 5 q2 + 20 q3 <20 && 512 q1 ^ 2 + 80 q1 (8-11 q2 + 4 q3) +25 (5 q2 ^ 2 + 16 q2 (2 + q3) +64 (-1 + q3) (1 + 2 q3)) <0
```

The command

```
To integrate[Boole[constraint3], {q1,0,5}, {q2,0,4}, {q3,0,1}]/ (10/3)
```

then, interestingly, it produces the "PPT probability" of Hilbert-Schmidt that the partial transposition of $ rho_3 $ it is positive defined,

```
(1/13720) (- 4312 + 5145 [Pi] + 2240 Sqrt[7] ArcCos[11/(8 Sqrt[2])]- 5160 Sqrt[7] ArcSin[(5Sqrt[(5Sqrt[(5Sqrt[(5Sqrt[7])/sixteen]- 6860 ArcTan[7] + 6280 Sqrt[7] ArcTan[(5Sqrt[(5Sqrt[(5Sqrt[(5Sqrt[7]) / 9])
```

which is approximately 0.461554. (This result was published as a comment in my previous query https://quantumcomputing.stackexchange.com/posts/5943/edit).

Now, I would like to solve similarly the even more formidable $ d = 4 $ problem. Then, the restriction (found by the execution of the positive definition of the sixteen nested minors of both $ rho_4 $ and its partial transposition $ rho_4 ^ {PT} $) take the form

```
restriction 4 = q1> 0 && q2> 0 && q3> 0 && q4> 0 && 5 q1 + 11 (q2 + 5 (q3 + q4)) <55 && 3375 q1 ^ 2 + 121 (7 q2 ^ 2 + 90 q2 (1 + q3-q4) +225 (1 +3 q3-q4) (-1 + q3 + q4)) <330 q1 (19 q2-15 (1 + q3-q4)) && (45 q1 + 11 (15 -7 q2-15 q3 + 45 q4)) (75 q1-11 (15 + q2-15 q3 + 45 q4)) <0
```

Execution of the order

```
To integrate[Boole[constraint4], {q3,0,1}, {q2,0,5}, {q1,0,11}, {q4,0,0,1}]/ (55/24)
```

would then give the corresponding probability of Hilbert-Schmidt PPT. (The GenericCylindricalDecomposition command suggested the particular ordering of the four variables for the 24 possible arrangements, but, of course, variations can be investigated).

Currently, using simply the free form of WolframCloud, my different attempts to perform the integration, by one method or another, are exhausted. In any case, the problem may be too formidable, by any means. (Maybe some transformations of variables could be effective).

Given these probabilities of PPT, the next question that would arise, of a nature that has never been addressed in a meaningful way, is how probabilities are divided between "entangled liaison" and "separable" states (see Fig. 3 of the appointment). Hiesmayr / Löffler paper).

This code can be used to generate $ rho_4 $

d = 4; W[k_, l_] : =

Sum[Exp[2 Pi I k n/d] Exterior[Times, S[n], S[Mod[n + l, 4]]]{n, 0,

d – 1}];

S[0] = {1, 0, 0, 0}; S[1] = {0, 1, 0, 0}; S[2] = {0, 0, 1, 0};

S[3] = {0, 0, 0, 1};

Omega[0, 0] = (1 / Sqrt[4]) Sum[

TensorProduct[S[s], S[s]], {s, 0, d – 1}]; Do[

Omega1[k, l] =

ArrayReshape[

TensorProduct[W[k, l], Identity matrix[4] Omega[0, 0]], {16, 16}]/

Sqrt[4], {k, 0, d – 1}, {l, 0, d – 1}]; Do[

P[k, l] =

Exterior[Times, Omega1[k, l].ConjugateTranspose[Omega1[k, l]]], {k, 0,

d – 1}, {l, 0, d – 1}]; den =

Sum[c[k, l] P[k, l], {k, 0, d – 1}, {l, 0, d – 1}]rho[d_] : = (1 – q1 / (d ^ 2 – (d + 1)) – q2 / (d + 1) –

q3 – (d – 3) q4) IdentityMatrix[d^2]/ d ^ 2 +

q1 P[0, 0]/ (d ^ 2 – (d + 1)) +

q2 / ((d + 1) (d – 1)) Sum[P[i, 0], {i, 1, d – 1}]+ (q3 / d) Sum[

P[i, 1], {i, 0, d – 1}]+ (q4 / d) Sum[Sum[P[i,z], {i, 0, d-1}]{z, 2, d-2}]rho[4]

The partial transposition of $ rho_4 $ you get by

```
ArrayFlatten[Transpose[Divide[rho[Transpose[Partition[rho[Transponer[Dividir[rho[Transpose[Partition[rho[4], {4,4}]]];
```