Definition: For $0\leq p\leq 1$, a $p$-trigadget is a pair of boxes, each with three input buttons (say, $X,Y,Z$, only one of which can be pressed) and an light which can flash one of two colors (blue or red, say), such that the light flashes as soon as a button is pressed, subject to the following constraints:
when the same button is pressed on both boxes, the lights on both boxes will always flash the same color, and probability $\frac{1}{2}$ for each (i.e., we get (blue,blue) with probability $\frac{1}{2}$ and (red,red) with probability $\frac{1}{2}$),
when different buttons are pressed on both boxes, the lights on both boxes will be the same with probability $1-p$ and different with probability $p$, with no bias toward one particular color (i.e., we get (blue,blue) and (red,red) with probability $\frac{1-p}{2}$ each, and (blue,red) and (red,blue) with probability $\frac{p}{2}$ each).
Remarks (connecting these $p$-trigadgets to more standard notions in information theory, as presented, e.g., in this survey):
It is possibly more standard to flip the light colors of one of the boxes in the definition (so that the boxes always flash a different color when the same button is pressed). I prefer the above presentation which is more symmetric, but this is, of course, immaterial. (I just mention this because one may be confused in trying to apply the standard inequalities to the gadgets I defined.)
If we allow instantaneous communication between the two boxes, the $p$-trigadget can easily be realized as follows: the first box whose button is pressed flashes either blue or red with probability $\frac{1}{2}$, and the second flashes the same color if the same button is pressed as the first box, and, if a different button is pressed it flashes the same color with probability $1-p$ and the other color with probability $p$. Of course, the whole issue is that we don't want to allow instantaneous communication (except insofar as it is posulated in the existence of such boxes).
Without allowing for instantaneous communication (i.e., under “local realism”), a $\frac{2}{3}$-trigadget can be realized as follows: the box pair has a hidden variable which is one of the six possible nonconstant maps $\{X,Y,Z\} \to \{\text{blue},\text{red}\}$, chosen equiprobably a priori, and the box flashes the image by that map of the button that was pressed. But it is an immediate consequence of the CHSH inequality that no $p$-trigadget for $p>\frac{2}{3}$ can be realized under local realism.
Allowing for a shared quantum state prepared in advance between both boxes, a $\frac{3}{4}$-trigadget can be realized, but no $p$-trigadget for $p>\frac{3}{4}$. (This is a fairly standard fact: the positive statement is easily shown using an EPR pair, and the negative statement follows from Tsirelson's bound.)
A $1$-trigadget is equivalent to a PR-box-pair (“PR” = “Popescu-Rohrlich”) in the following sense: a PR-box-pair is the same as a $1$-trigadget in which only the $X$ and $Y$ buttons are available in one box and only the $X$ and $Z$ buttons are available in the other (this can be taken as a definition of a PR-box-pair), and conversely, a $1$-trigadget can easily be realized with three PR-box-pairs.
A $p$-trigadget can simulate a $p'$-trigadget for any $0\leq p'\leq p$: just randomly choose between a $p$-trigadget with probability $p'/p$ or a $0$-trigadget (which is trivial to make) with probability $(p-p')/p$). So, the greater the $p$, the more powerful the $p$-trigadget.
To summarize, $\frac{2}{3}$-trigadgets are doable, $\frac{3}{4}$-trigadgets (but no more) can be made using shared quantum states, and $1$-trigadgets are extreme nonlocal but nonsignalling. These correspond to the best we can do using well-studied correlation sets. What I would like to know is if anything can be said about $p$-trigadgets for other values of $p$.
Broad question: Has the notion of $p$-trigadget for $\frac{2}{3}<p<1$ and $p\neq\frac{3}{4}$, or something equivalent to it, or the set of correlations that they enable, been studied somewhere before?
Specific question: Is the hierarchy of $p$-trigadgets strict? In other words, is it true that if $p<p'$ we cannot realize a $p'$-trigadget using $p$-trigadgets? More precisely:
- Assume $p'>p$: if Alice and Bob have an unlimited supply of $p$-trigadgets, but — after initial concertation — cannot communicate (i.e., are bound by local realism) outside of the use of these $p$-trigadgets, is it true that they cannot simulate a $p'$-trigadget (with Alice playing the role of one box and Bob the box of the other)?
(There should also be an analogous question with $\frac{3}{4} \leq p < p'$ further assuming that Alice and Bob are also allowed to prepare shared quantum states, but I'm not sure how to even state the question properly.)
Note: I am concentrating on “tri”gadgets with two colors here for simplicity, but more generally, for any finite graph $G$, set of $n$ colors and $0\leq p\leq 1$, there is a generalized gadget consisting of two boxes with the vertices of $G$ as buttons, such that if the same button is pressed on both vertices, the same color flashes, and if different colors are pressed, different colors flash with probability $p$. I had asked about the largest $p$ such that such a generalized gadget can be realized in this question. The “trigadget” case presented above corresponds to $G$ being the triangle graph and $n=2$, i.e., the smallest nontrivial case. (I hope this helps motivate the question.)
