I've determined a number of probabilities pertaining to three subsets of the "magic simplex" of quantum states described in https://arxiv.org/abs/1212.5046, and certainly would like to have a pictorial representation of their interrelations--presumably, in the form of a Venn diagram (cf. Fig. 3 in cited paper).

One set--$A$, call it--is composed of those states, the density matrices for which have "positive-partial-transposes" (PPT). Its probability is $\frac{8 \pi}{27 \sqrt{3}} \approx 0.537422$.

Another set--$B$--consists of those states that pass a certain (mutually-orthogonal-bases [MUB]) test for entanglement. Its probability is $\frac{1}{6} \approx 0.16667$.

The third set--$C$--consists of states that pass another (Choi witness) test for entanglement. Its probability is also $\frac{1}{6} \approx 0.16667$.

The intersections of $A$ and $B$ and of $A$ and $C$ give conceptually-important "bound-entanglement" probabilities. Both amounts are $-\frac{4}{9}+\frac{4 \pi }{27 \sqrt{3}}+\frac{\log (27)}{18} \approx 0.00736862$.

$B \land C$ is $\frac{1}{9} \approx 0.11111$.

$B \lor C$ is $\frac{2}{9} \approx 0.22222$.

Both $\neg B \land C$ and $B \land \neg C$ are $\frac{1}{18}$.

$A\land \neg B\land \neg C$ gives $\frac{1}{9} (8 - \log{3}) \approx 0.52268$.

$A \land B \land C$ is void.

So, I would like a (planar?) Venn-type diagram representing--as well as possible--the intersection and union relations between $A, B$ and $C$ and the larger set $D$ of probability 1, of which they are subsets.

An immediate idea would be to try to represent them by circles--but, I think, there is also an approach ("Euler diagrams" https://en.wikipedia.org/wiki/Euler_diagram) in which rectangles are employed.


closed as unclear what you're asking by Roman, MarcoB, corey979, m_goldberg, Alex Trounev May 10 at 19:02

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    $\begingroup$ It would be great if you could add some code to your question to help those of us with less maths knowledge and more mathematica knowledge understand what you're asking. $\endgroup$ – Carl Lange May 9 at 19:23

Here is an attempt to represent the relations between the three subsets of the larger set $D$ of probability 1, represented by a circle ($D$) centered at the origin $\{0,0\}$ of radius $\frac{1}{\sqrt{\pi}}$.

I used the circle-circle intersection formula for "the area of a general asymmetric lens obtained from circles of radii R and r and offset d" (http://mathworld.wolfram.com/Lens.html)

-(1/2) Sqrt[(d + r - R) (d - r + R) (-d + r + R) (d + r + R)] + r^2 ArcCos[(d^2 + r^2 - R^2)/(2 d r)] + R^2 ArcCos[(d^2 - r^2 + R^2)/(2 d R)]

with the three radii

rA = Sqrt[(8 Pi/(27 Sqrt[3]))/Pi]; rB = Sqrt[1/6]; rC = Sqrt[1/6];

and the three overlap probabilities

overlapAC = -(4/9) + (4 \[Pi])/(27 Sqrt[3]) + Log[27]/18; overlapAB =  -(4/9) + (4 \[Pi])/(27 Sqrt[3]) + Log[27]/18; overlapBC = 1/9

The command

Solve[{inter[rA, rB, dAB] == overlapAB,inter[rA, rC, dAC] == overlapAC, inter[rB, rC, dBC]; == overlapBC}, {dAB, dAC, dBC}]

was not successful, and one could use FindRoot.

However, I did (am doing) a random search within $D$, constraining the possible locations for the center of $A$ to lie within a circle--again centered at the origin--of radius $1-\frac{8 \pi}{27 \sqrt{3}}$, in order that it not extend outside the overall (probability 1) set $D$. Similarly, the origins of the circles representing $B$ and $C$ have to lie within circles of radii $\frac{5}{6}$.

I tried minimizing a function (many possible candidates) of the fits of the overlap areas to the given probabilities.

The result is given by the webpage

Venn-diagram attempt

I would appreciate comments as to Mathematica commands for coloring/labeling ,...the sets so as to highlight the areas of intersection--or any alternative approaches to addressing the underlying question.


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