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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?

glS
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Paul B. Slater
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1 Answers1

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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.

AHusain
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