So not having yet read through OP I am not terribly surprised that this is true and I can kind of give a quick sketch in terms of a QM game that I want everyone to know, called Betrayal.
The idea is that it's a collaborative game for three people, you are trying to work together to beat the rules of the game. Meanwhile the rules are trying to set you up so that one of the people betrays the other two. In 3 relativistically separated rooms (so they can’t communicate) they go, where they find a screen and buttons labeled 0 and 1. The screen displays a prompt, each teammate presses exactly one of the buttons once before time runs out, then the three numbers pressed get summed together into a number.
25% of the time we run a “control round,” everyone gets a prompt to make the sum of their numbers even, and they win if the sum is even. The easiest way is if everyone hits 0, 0+0+0 is even. But a team can also answer 0+1+1 or so and win. Otherwise we randomly choose one to be the traitor and send them the control prompt, to make the sum even. But we send the other two the prompt to make the sum odd! In this case the team will only win if their joint sum is odd.
Long story short, classical players of this game have a success probability bounded from above by 75%. This is the Bell inequality. But quantum capable players can walk in with a GHZ state,
|+++> + |–––>,
which only collapses to even sums. If they have to do a control round they will all just measure this in the computational basis.
The more interesting thing, where i really matters, comes during the traitor rounds. Here you want to perform the phase rotation gate in the Hadamard basis,
|+> → |+>,
|–> → i |–>,
And any two of them can thereby switch the state to
|+++> – |–––>,
a state which only has odd configurations. Quantum players can win 100% of the time. Over multiple independent trials you should be able to observe the inequality violations even if quantum coherence were to limit your success probability to 90%.
I suspect that the inequality here is something similar, quantum mechanics but you can only form real-coefficient superpositions, and therefore you cannot take the square root of a unitary transformation just by doing it for half the time, per Schrödinger.
Yes, the (very nice) game you described gives a separation between classical and quantum mechanics. However, there is a strategy in real quantum mechanics which also achieves a 100% winrate for the players (you just need higher dimensional Hilbert spaces for each player).
Instead of preparing |+++> + |–-->, you prepare the state (|+++>|x> + |–-->|x>) Here, |x> = |000>-|011>-|101>-|110>, and one qubit is sent to each player.
That is, you give each of Alice, Bob and Charlie an extra qubit. They can now measure in the computational basis on both qubits. And in the betrayal round two of the players can perform the orthogonal transformation id \otimes J, (controlled on having |->) where J = {{0,-1},{1,0}}. You can check that whenever exactly two players perform this operation on their systems you get back the state (|+++>-|--->)|x>, and thus your previous strategy works.
This simulation strategy for any full multipartite causal structure is described in arXiv:0810.1923. What OP has shown (roughly) is that three players connected as in A <-> B <-> C (where <-> is some shared randomness or quantum state) then this simulation breaks, and indeed there is a gap between what you can achieve in real and complex quantum mechanics.
So GHZ is just a name for a specific arrangement that is maximally entangled.
John Preskill has some lectures on quantum computing for the University of Waterloo I believe, also a Hans Bethe lecture at Cornell. If you are just looking for an hour's commitment to understand a little better, I would go with one of those.
If you wanted an actual textbook, Nielsen and Chuang is very popular... The other place I would look would be OpenCourseWare, you might be able to find some good problems to work on there. Sometimes video lectures can be good if you can pause the video right after a problem was introduced and try to solve it yourself before you get the answer from the professor.
The difficulty in being an autodidact is, listening to stories around a campfire is deep in our bones, it makes us feel good. But it's not a very efficient way to learn. So there is a mismatch where watching a TED talk feels like you have just changed everything, but then if I come to you a month later probably nothing has changed.
Text is a lot faster, as a medium. Way slower to write but seekable, skimmable, can contain links to previous sections... I'm pretty sure we also retain more of it. But that's not the main problem with videos/TED talks. Like, the text form of TED talks is someone telling you how monads are burritos and that makes it all better.
It's too clean?
Good learning is messy. A good abstraction allows you to clean up a mess of confusion in your head. This confused mess can only exist if you have created it. So you have to do lots of examples, exercises, memorize strange times when you have been wrong about things and your expectations don't align with the problem domain... If you think about learning a language, there is that phase where you don't know which thing to use when and your words are all out of order in the sentence... Per Ira Glass the only way to improve is to do lots of work, put yourself on a schedule, grind through mediocrity. The TED talk/monad tutorial fallacy is that we can give our children an easier time than we had it. It's BS. We can't. “I made so many mistakes, I will help you so that you don't have to deal with that pain” blithely unaware that the pain was how you learned it, that learning is pain.
Sorry, didn't mean to rant and now it seems awkward to delete it.
The idea is that it's a collaborative game for three people, you are trying to work together to beat the rules of the game. Meanwhile the rules are trying to set you up so that one of the people betrays the other two. In 3 relativistically separated rooms (so they can’t communicate) they go, where they find a screen and buttons labeled 0 and 1. The screen displays a prompt, each teammate presses exactly one of the buttons once before time runs out, then the three numbers pressed get summed together into a number.
25% of the time we run a “control round,” everyone gets a prompt to make the sum of their numbers even, and they win if the sum is even. The easiest way is if everyone hits 0, 0+0+0 is even. But a team can also answer 0+1+1 or so and win. Otherwise we randomly choose one to be the traitor and send them the control prompt, to make the sum even. But we send the other two the prompt to make the sum odd! In this case the team will only win if their joint sum is odd.
Long story short, classical players of this game have a success probability bounded from above by 75%. This is the Bell inequality. But quantum capable players can walk in with a GHZ state,
which only collapses to even sums. If they have to do a control round they will all just measure this in the computational basis.The more interesting thing, where i really matters, comes during the traitor rounds. Here you want to perform the phase rotation gate in the Hadamard basis,
And any two of them can thereby switch the state to a state which only has odd configurations. Quantum players can win 100% of the time. Over multiple independent trials you should be able to observe the inequality violations even if quantum coherence were to limit your success probability to 90%.I suspect that the inequality here is something similar, quantum mechanics but you can only form real-coefficient superpositions, and therefore you cannot take the square root of a unitary transformation just by doing it for half the time, per Schrödinger.