9 ms·

> nugget of actual math If you are seriously looking for a nugget I would suggest trying to understand the nonlocal game example (which was hidden in the write

by dfdz 7y ago

> nugget of actual math If you are seriously looking for a nugget I would suggest trying to understand the nonlocal game example (which was hidden in the write up) "But first, to see how the games work, let’s imagine two players, Alice and Bob, and a 3-by-3 grid. A referee assigns Alice a row and tells her to enter a 0 or a 1 in each box so that the digits sum to an odd number. Bob gets a column and has to fill it out so that it sums to an even number. They win if they put the same number in the one place her row and his column overlap. They’re not allowed to communicate. Under normal circumstances, the best they can do is win 89% of the time. But under quantum circumstances, they can do better. Imagine Alice and Bob split a pair of entangled particles. They perform measurements on their respective particles and use the results to dictate whether to write 1 or 0 in each box. Because the particles are entangled, the results of their measurements are going to be correlated, which means their answers will correlate as well — meaning they can win the game 100% of the time." These notes might help to understand the basic idea: https://www.scottaaronson.com/qclec/14.pdf https://www.scottaaronson.com/qclec/14.pdf In particular, after understanding why the best they can is win 89% on normal circumstances, the easier examples at the beginning of the notes provide intuition of how a quantum strategy might help for the 3-by-3 grid game. Edit: also see the write up by ahelwer below.

ahelwer 7y ago
I wrote up an in-depth introduction to a different nonlocal game (CHSH) here: https://ahelwer.ca/post/2018-12-07-chsh/ https://ahelwer.ca/post/2018-12-07-chsh/ It's a bit more complicated to follow but absolutely worth the work.
taneq 7y ago
> Imagine Alice and Bob split a pair of entangled particles. They perform measurements on their respective particles and use the results to dictate whether to write 1 or 0 in each box. Why not just imagine Alice tells Bob how she's going to choose what to write in the boxes? That works 100% of the time too. I don't see a distinction between "Alice and Bob communicate" and "a pair of entangled particles gives Alice and Bob a piece of shared information that, by prior arrangement, they both interpret in the same way."
- cyphar 7y ago
  Most of Bell's thought-experiments result in fairly unintuitive answers, so it's quite understandable that the distinction would be unclear to someone who hadn't seen this problem before. The description of the game in the article is incredibly simplified. The original Bell game is that you take two particles and separate them spacially so that any local (slower-than-light) communicaton cannot interfere with the experiment. You then measure their spins in two orientations (the absolute orientation isn't important, only the angle between the two measurements is important) and repeat a large number of times. Then, sum how many times the two measurements had the same spin (+1) or opposite spins (-1) and divide it by the number of measurements to get the "correlation" (scare quotes because this is not the same as what statisticians mean when they say "correlation"). Repeat for a large number of different relative angles and plot "correlation" vs angle. Quantum mechanics predicts (and experimental data produces) an inverse cosine "correlation" curve. However, it is simply not possible to produce that "correlation" curve using a local, hidden variable theory of quantum mechanics. Why? With some slight hand-waving, it's because the two particles don't know along which (relative) angle the other particle will be measured ahead of time -- if you permit non-locality (faster-than-light communication or "spooky" action at a distance) then this problem goes away. Now, the proof that this is the case is far more involved than this (and to be honest I'm not sure I understand it well enough to explain it). But hopefully that gives you some idea why the "just use an RNG" method is not sufficient.
  - axilmar 7y ago
    > it's because the two particles don't know along which (relative) angle the other particle will be measured ahead of time Since the two particles are created from the same source, why couldn't it be that the two particles are affected by the initial state of their source so as that to appear correlated?
    - cyphar 7y ago
      The nature of the source is the reason they are correlated in that particular way (that is effectively what entanglement is achieving, at least in this simple example). In particular, the source in the classic Bell inequalities produced particles that had opposite spins (guaranteed by conservation of angular momentum). But that's effectively just a description of the problem. It doesn't really help you come up with a local hidden-variable theory of QM could produce the same "correlation" vs angle curve. The problem (in computer science terms) is that you need to come up with an algorithm (which for a given particle (A or B) takes some state (called lA or lB), and an angle parameter (oA and oB)) that gives you an output which has the statistical distribution which agrees with QM if you randomly sample lA and lB values. In other words, write an algorithm such that the "correlation" of Measure(oA, lA) and Measure(oB, lB) is always equal to precisely -cos(|oA - oB|) without using global variables. It turns out this is (provably) impossible. NOTE: In the real experiment you're actually measuring in 3D, so the angles should actually be unit vectors and the angle difference is actually (a function of) the dot product.
p0ckets 7y ago
Thank you. The game description in the article didn't really make sense since it skips the (impossible) claim that: 1. Every row has an even sum 2. Every column has an odd sum

by dfdz 7y ago

ahelwer 7y ago

I wrote up an in-depth introduction to a different nonlocal game (CHSH) here: https://ahelwer.ca/post/2018-12-07-chsh/ https://ahelwer.ca/post/2018-12-07-chsh/ It's a bit more complicated to follow but absolutely worth the work.

taneq 7y ago

> Imagine Alice and Bob split a pair of entangled particles. They perform measurements on their respective particles and use the results to dictate whether to write 1 or 0 in each box. Why not just imagine Alice tells Bob how she's going to choose what to write in the boxes? That works 100% of the time too. I don't see a distinction between "Alice and Bob communicate" and "a pair of entangled particles gives Alice and Bob a piece of shared information that, by prior arrangement, they both interpret in the same way."

cyphar 7y ago
Most of Bell's thought-experiments result in fairly unintuitive answers, so it's quite understandable that the distinction would be unclear to someone who hadn't seen this problem before. The description of the game in the article is incredibly simplified. The original Bell game is that you take two particles and separate them spacially so that any local (slower-than-light) communicaton cannot interfere with the experiment. You then measure their spins in two orientations (the absolute orientation isn't important, only the angle between the two measurements is important) and repeat a large number of times. Then, sum how many times the two measurements had the same spin (+1) or opposite spins (-1) and divide it by the number of measurements to get the "correlation" (scare quotes because this is not the same as what statisticians mean when they say "correlation"). Repeat for a large number of different relative angles and plot "correlation" vs angle. Quantum mechanics predicts (and experimental data produces) an inverse cosine "correlation" curve. However, it is simply not possible to produce that "correlation" curve using a local, hidden variable theory of quantum mechanics. Why? With some slight hand-waving, it's because the two particles don't know along which (relative) angle the other particle will be measured ahead of time -- if you permit non-locality (faster-than-light communication or "spooky" action at a distance) then this problem goes away. Now, the proof that this is the case is far more involved than this (and to be honest I'm not sure I understand it well enough to explain it). But hopefully that gives you some idea why the "just use an RNG" method is not sufficient.
- axilmar 7y ago
  > it's because the two particles don't know along which (relative) angle the other particle will be measured ahead of time Since the two particles are created from the same source, why couldn't it be that the two particles are affected by the initial state of their source so as that to appear correlated?
  - cyphar 7y ago
    The nature of the source is the reason they are correlated in that particular way (that is effectively what entanglement is achieving, at least in this simple example). In particular, the source in the classic Bell inequalities produced particles that had opposite spins (guaranteed by conservation of angular momentum). But that's effectively just a description of the problem. It doesn't really help you come up with a local hidden-variable theory of QM could produce the same "correlation" vs angle curve. The problem (in computer science terms) is that you need to come up with an algorithm (which for a given particle (A or B) takes some state (called lA or lB), and an angle parameter (oA and oB)) that gives you an output which has the statistical distribution which agrees with QM if you randomly sample lA and lB values. In other words, write an algorithm such that the "correlation" of Measure(oA, lA) and Measure(oB, lB) is always equal to precisely -cos(|oA - oB|) without using global variables. It turns out this is (provably) impossible. NOTE: In the real experiment you're actually measuring in 3D, so the angles should actually be unit vectors and the angle difference is actually (a function of) the dot product.

p0ckets 7y ago

Thank you. The game description in the article didn't really make sense since it skips the (impossible) claim that: 1. Every row has an even sum 2. Every column has an odd sum