Thread by @gro_tsen

Because of the latest Nobel price announcement, maybe it's time I should dig up some threads I wrote two years ago, and explain some things that quantum mechanics does and doesn't let you do with and around the curious notion of “p-gadgets”. 🧵🔽 •1/27

So, given a real number 0≤p≤1 (the cases p=½, p=¾ and p=1 will most concern us), a “p-gadget” is a hypothetical apparatus consisting of a pair of devices, which I will call “thingies”. Thingies always come in pairs: the gadget is the pair of twin thingies. •2/27

Each thingy has 3 buttons on them, “X”, “Y” and “Z”. When you press one of the buttons, a light on the thingy flashes (instantly), either blue for “yes” or red for “no”. That's all a thingy does. By itself, it's pretty useless. Furthermore, it can only be used once! •3/27

So all you can do with a thingy is press one of the X, Y or Z buttons, get a yes/no answer, and then it becomes junk. But there are rules connecting how the twin thingies in a gadget behave! Bear with me, because that's the whole point of the gadget. •4/27

The rules (below) specify (i.e., restrict) what answers (yes or no) the two twin thingies of a given pair can give you. The rules apply regardless of how, when or where the thingies are used. Crucially, the thingies do not communicate with each other when they are used. •5/27

The rules (on the answers given by both thingies for a given p-gadget) are: ① if you press the SAME button on both thingies in the pair, you ALWAYS get the same answer from both, but ② if you press DIFFERENT buttons, you have probability ≥p of getting DIFFERENT answers. •6/27

That's it! That's all that p-gadgets guarantee you. (Note that, in ②, the probability is understood for any setting of the buttons, even an adversarial one, across many gadgets. For p=1, interpret “probability 1” as “always” for simplicity.) •7/27

These gadgets might not seem terribly useful, but let's skip over this for the moment and ask: can they be made? Clearly, the greater the value of p, the more difficult / powerful (i.e., the more constrained) the gadget. Which can be made, and how? •8/27

Well, it's very easy to make a (½)-gadget (i.e. p=½: in ②, different buttons settings must have ≥½ chance of giving different answers): to make such a gadget, simply flip three fair coins, and choose the answers to be given for X, Y and Z by both gadgets thusly. •9/27

Namely, the yes/no answer for each of the X, Y and Z settings is chosen independently with probability ½ for each of “yes” or “no”; it is chosen at the gadget's creation, and it will be the same for both thingies in the pair (this guarantees ①). •10/27

If different buttons are pressed, we get the result of two different coin flips, which have probability ½ of being different, so ② is satisfied with p=½, as announced: so we have (trivially) made a (½)-gadget (factory), in a classical world. •11/27

This gadget (described in tweets 9–11) is a classical one, and it uses “hidden variables”: both thingies embed the answer they will give when each button is pressed, and they are hidden because they can be revealed only by pressing the corresponding button. •12/27

Now if you think about it a bit more, you see that in a “hidden variables” perspective you can't do better than p=½: condition ① imposes that both thingies must have the same predefined yes-or-no answer for X, Y and Z, … •13/27

… but it's an easy exercise in probabilities to see that we can't find a probability distribution on the 8 possible combinations of all three so that the probability of two different values being distinct is ≥p, for p>½. So a (½)-gadget is the best we can make thusly. •14/27

Now of course we might try something different. If the thingies in a pair could communicate, it would be easy to make a 1-gadget: the first thingy to be activated always answers “yes”, say, and communicates the button pressed to the other, … •15/27

… which then will give the same “yes” answer for the same button, and “no” for the other. This would be a 1-gadget, but we can't do that because the thingies can't communicate: they must work even when separated by distances so huge light cannot get across. •16/27

I think it's really worth taking a moment to ponder how one might proceed to make a p-gadget for p>½, even a 1-gagdet, and what one might do with such things, because it takes a while to get used to the concept!

Let me now describe what is known about making p-gadgets.
•17/27

‣ Classically, you can make a (½)-gadget, as I explained, but you CANNOT make a p-gadget for p>½.

(Of course, there are assumptions here — “local realism” — that I glossed over, e.g., that you cannot predict in advance which button will be pressed.)
•18/27

‣ Using quantum mechanics, you CAN make a (¾)-gadget, but you CANNOT make a p-gadget for p>¾.

Both parts of this are remarkable! Quantum mechanics lets you go beyond the p=½ limit of local realism, but not arbitrarily far either!
•19/27

(Essentially, to make a (¾)-gadget using QM, if you know what this means, you start with two entangled spin-½ particles with state (|↑↓⟩−|↓↑⟩)/√2, you put one in each thingy, and each X,Y,Z button measures its spin on an axis separated at 2π/3 from the other.) •20/27

Note that this HAS been done and tested experimentally. This is what the whole fuss is about. Conceptually, it's an easy experiment, but actually eliminating all “loopholes” to eliminate possible local hidden variables explanations is experimentally quite challenging. •21/27

The UPPER bound on what we can do with quantum mechanics is equally remarkable. Basically this is “Tsirelson's bound”, but there are lots of subtle convex inequalities at play which I don't claim to fully understand (and I've forgotten part of what little I knew). •22/27

‣ Even a 1-gadget WILL NOT allow you to communicate faster than light, but it WILL allow you to communicate far more efficiently (it collapses communication complexity).

⇒ We can conceive gadgets beyond what QM allows, and they too can be studied (mathematically).
•23/27

(Actually, the standard term here is a PR-box or Popescu-Rohrlich box, which is a bit different — less symmetrical — from the 1-gadget I described, but the two are equivalent in the sense that we can make either of these things from the other.) •24/27

If you want to know more about all this, there are a few past threads of mine when I was trying to learn about such stuff, and some more knowledgeable people gave me interesting answers and pointers, such as this thread 🔽 and the ones it links to. •25/27

https://twitter.com/gro_tsen/status/1273632476578152449

I had also written a (part of a) blog post on the subject. It doesn't say much (beyond what I wrote in this thread) and it's in French, but there are links to a number of papers some of which are a great introduction to the subject. •26/27 madore.org/~david/weblog/…

Probably the best place to start is the nicely written survey paper “Bell nonlocality” by Brunner, Cavalcanti, Pironio &al, which is introductory and covers various aspects of the question, from mathematics to experiment. •27/27 arxiv.org/abs/1303.2849

PS: For mathematically precise definitions of various convex sets of possible correlations (“local”, “non-signalling”, and 3 “quantum” sets whose relations are linked to the Connes embedding conjecture), see §2 of Dykema, Paulsen & Prakash: arxiv.org/abs/1709.05032 •28/(27+1)

PS2: Maybe I should also link this MathOverflow question where I define related concepts of “p-coloring” and “quantum p-coloring” a graph for 0≤p≤1; the above says that the triangle graph is ½-colorable and quantum ¾-colorable with 2 colors. mathoverflow.net/q/365464/17064 •29/(28+1)

⚠️ CORRECTION: I made a silly mistake, and every “½” in the thread above this tweet should be changed to “⅔”. The thread quoted below 🔽 explains this in detail (and gives proofs of all relevant statements). Sorry about the confusion! •30/(29+1)

https://twitter.com/gro_tsen/status/1578047461029036040