# What are the ranges of the four $q$ parameters in the magic simplex of Bell states formula?

Equation (7) in the 2012 paper, "Complementarity Reveals Bound Entanglement of Two Twisted Photons" of B. C. Hiesmayr and W. Löffler for a state $$\rho_d$$ in the "magic simplex" of Bell states $$\begin{equation} \rho_d= \frac{q_4 (1-\delta (d-3)) \sum _{z=2}^{d-2} \left(\sum _{i=0}^{d-1} P_{i,z}\right)}{d}+\frac{q_2 \sum _{i=1}^{d-1} P_{i,0}}{(d-1) (d+1)}+\frac{q_3 \sum _{i=0}^{d-1} P_{i,1}}{d}+\frac{\left(-\frac{q_1}{d^2-d-1}-\frac{q_2}{d+1}-(d-3) q_4-q_3+1\right) \text{IdentityMatrix}\left[d^2\right]}{d^2}+\frac{q_1 P_{0,0}}{d^2-d-1} \end{equation}$$ yields "for $$d=3$$ the one-parameter Horodecki-state, the first found bound entangled state".

No explicit ranges — in which I am interested — for the four $$q$$ parameters are given, though. Any thoughts/insights?

It looks like the only relation they say is $$\text{IdentityMatrix}[3^2]=\sum P_{k,l}$$.

You get a linear combination of $$P_{k,l}$$. Those are vertices of a $$d^2-1$$ simplex so the coefficents $$c_{k,l}$$ are baryocentric coordinates.

You can then match with the previous more general definition of $$\rho_d$$ term by term on each of the $$c_{k,l}$$.

The inequalities $$0 \leq c_{k,l} \leq 1$$ turn into $$2d^2$$ inequalities on the $$q_i$$.

It looks like that how the $$q_i$$ were constructed in the first place.

• For the $d=3$ case, the desired range conditions appear to be $0<q_1<5, 0<q_2<8, 0<q_3<3, q_1/5+q_2/8+q_3/3<1$. Utilizing this information, I have obtained for the Hilbert-Schmidt PPT-probability of the states $\rho_d$, as given in the formulation of the question, the result $-\frac{11}{35}+\frac{3 \pi }{8}-\frac{129 \sin ^{-1}\left(\frac{5 \sqrt{7}}{16}\right)}{49 \sqrt{7}}+\frac{8 \cos ^{-1}\left(\frac{11}{8 \sqrt{2}}\right)}{7 \sqrt{7}}-\frac{1}{2} \tan ^{-1}(7)+\frac{157 \tan ^{-1}\left(\frac{5 \sqrt{7}}{9}\right)}{49 \sqrt{7}} \approx 0.461554$. I will now try the $d=4$ case. Apr 20, 2019 at 14:35
• Second thoughts about the above formula. Apr 20, 2019 at 18:37
• Yes the formula does hold--using the range constraint $q_1>0, q_2>0, q_3>0, 0<1-q_1/5-q_2/4-q_3<1$. Apr 21, 2019 at 0:18
• @PaulB.Slater I've edited the formatting in your comments a bit. Apr 21, 2019 at 7:19