To begin, we have two constraints \begin{equation} C1=x>0\land z>0\land y>0\land x+2 y+3 z<1 \end{equation} and \begin{equation} C2=x>0\land y>0\land x+2 y+3 z<1\land x^2+x (3 z-2 y)+(y+3 z)^2<3 z. \end{equation} $C1$ ensures the nonnegative-definiteness of a class of $9 \times 9$ ("two-qutrit") density matrices ($\rho$).

$C2$ also ensures this, as well as the nonnegative-definiteness of the "partial transpose" ($\rho^{PT}$) of $\rho$.

The integration--subject to $C1$--of the value 36 over the unit cube $\{x,y,z\} \in [0,1]^3$ yields 1.

The integration--subject to $C2$--of the value 36 over the unit cube yields (the Hilbert-Schmidt positive partial transpose probability) $\frac{8 \pi}{27 \sqrt{3}} \approx 0.537422$.

Now, we are interested in similarly enforcing both $C1 \land C3$ (yielding an "entanglement probability") and $C2 \land C3$ (yielding a "bound-entanglement probability"), where, the entanglement constraint $C3$ is \begin{equation} b \left(-x \left(a^2-a (b+2)+2 b+1\right)+2 y (a (-a+b+2)+b-1)-3 z (a-b-1) (a+b-1)+(a-1)^2\right)<0, \end{equation} with its three parameters $a,b,c$ subject to \begin{equation} C4= b>0\land c>0\land 0<a<1\land a+b+c=2\land (a-1)^2=b c. \end{equation}

Now, I suspect these last two problems are too difficult to resolve in their full generality (leaving $a,b,c$ unspecified).

But, to begin, if we take \begin{equation} \{a,b,c\}=\left\{\frac{1}{4} \left(3-\sqrt{5}\right),\frac{1}{2},\frac{1}{4} \left(3+\sqrt{5}\right)\right\}, \end{equation} the integration of 36 over the unit cube subject to $C1 \land C3$ yields $\frac{5}{132} \left(5+\sqrt{5}\right) \approx 0.274093$.

Alternatively, for \begin{equation} \{a,b,c\} = \left\{\frac{1}{3},\frac{1}{3},\frac{4}{3}\right\}, \end{equation} the integration of 36 over the unit cube subject to $C1 \land C3$ yields $\frac{125}{486} \approx 0.257202$.

However, I have not been so far able to obtain the counterparts for these last two results for $C2 \land C3$. (Using numerical integration, we get the much lower values of 0.001497721920258410 and 0.003256122941383665, respectively.)

A set of values of $a,b,c$ which satisfy $C4$ are \begin{equation} \left\{\frac{2}{3} (\cos (\alpha )+1),\frac{2}{3} \left(-\frac{1}{2} \sqrt{3} \sin (\alpha )-\frac{\cos (\alpha )}{2}+1\right),\frac{2}{3} \left(\frac{1}{2} \sqrt{3} \sin (\alpha )-\frac{\cos (\alpha )}{2}+1\right)\right\}, \end{equation} for $\frac{\pi}{3} \leq \alpha \leq \frac{5 \pi}{3}$.

So, I would like to obtain results of integration of the value 36 over $[0,1]^3$ of $C1 \land C3$ and $C2 \land C3$ for either specific values of $a,b,c$, satisfying $C4$, or even without particular values being specified.