Is it possible to use quantum mechanics to create a gadget with the following “paradoxical” properties?

When used, the gadget creates a pair of thingies. On each thingy, one of three properties, A, B or C, can be measured. These are boolean (yes/no) values. …

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… But measuring any one of the three properties destroys the thingy in question. So you can only measure one of the three properties. However, remember that the thingies are created in pairs by the gadget: …

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‣ If the same property is measured on both thingies of the same pair (both A, or both B, or both C), they will ALWAYS AGREE.
‣ If different properties are measured on both thingies of the same pair (say, one A and one B), they will ALWAYS DISAGREE.

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So for example, if the gadget creates a pair (X,Y) of thingies, and I measure A for X and get a positive result, I know with certainty that measuring A for Y will give a positive result, while measuring B or C will give a negative result. •4/12

It's pretty obvious that this is classically impossible, i.e., impossible with “hidden variables”: if we assume each thingy has three hidden boolean variables (A,B,C) (of which we can only read one), then the two thingies in a pair have the same values for all, … •5/12

… and each of the three variables must differ from the other two, which is impossible for boolean variables. So, not possible. But is it possible using quantum mechanics? If so, how? If not, why not? •6/12

One classical Bell test setup, where two spin-½ particles are emitted with opposite spins and then measured using one of three axes (A,B,C) at 120° from one another (and an additional 180° between the two particles) will do something similar: … •7/12

… if the same measure (A,B,C) is performed on both particles, they always agree (spins are opposite, but I added an extra 180° so I'll say they agree), while if different measures are performed, they'll agree with probability ¼ and disagree with probability ¾. •8/12

This (¼,¾) is also classically ruled out, but less obviously so: one needs to compute a few inequalities on probabilities, to see that it's ruled out. Annoying. So I'd like to know if the more extreme version I described above can be realized quantumly. •9/12

If it CAN be realized quantumly, why is this experiment not used when illustrating the weirdness of the quantum world? It seems like it would be the ultimate example in quantum weirdness: elegant, very simple to describe, obviously paradoxical. •10/12

If it CAN'T be realized, quantumly, it means that while classically we can only get probability of disagreement of ½ for different measurements, quantumly we can get up to ¾ but not 1: so what's the upper bound and where do I find a proof of this upper bound? •11/12

And also: what kind of even-weirder-than-quantum type of laws of physics would be needed for my gadget to exist? I would expect discussions of Bell inequalities to explain this, but I can't find it. •12/12