Quantum mechanics is weird. You have probably heard this before. If you are a scientist you might have even said it before when trying to explain something to a layperson. If you are not a scientist then you have probably heard it and it has probably been frustrating to hear, because usually it is given in lieu of an explanation. It is both a true statement, and a dangerous one. Why is it dangerous? Because it gives people the wrong impression. If over and over again you hear "quantum mechanics is weird", as an answer to your questions, you begin to start hearing "quantum mechanics is magic". This sets a bad precedent for scientific literacy. I think most scientists understand this fact. Nevertheless, as a scientist it can often be difficult to avoid this phrase. One of the reasons is that you do not wish to bore people with math. There is also a deeper reason why this happens for quantum mechanics, and that is that quantum mechanics is weird in a particular way such that it is very hard to tell a story about what is going on in a quantum mechanical system, and stories are typically the medium of explanation.

Given that this blog is probably going to have a lot to say about quantum mechanics, it is very likely that we are going to be using this kind of phrase a lot. So I feel it is important to start things off with some concrete discussion about precisely what we mean when we say "quantum mechanics is weird". I am not going to talk about all the ways that it is weird, but I am going to talk about one of the most important ways, and that is how it informs our world-view. If you have always heard that quantum mechanics is weird, but never really understood in what way, then this is the post for you.

The most popular way of presenting this subject is by talking about the EPR paradox. However there is a challenge with the EPR paradox and that is that it involves Bell's inequality. Remember, we want to avoid boring people with math. So instead I am going to draw from an excellent talk given by Sidney Coleman in 1994 at Harvard. There he proposes an improved pedagogical example of what makes quantum mechanics so weird. I should say that this kind of thing has been blogged about before. Indeed it behooves me to mention Luboš Motl's blog post where I was introduced to the lecture by Sidney Coleman. My purpose here is not to present something new, or as my own, but to further the dissemination of these ideas.

The example that we are going to look at is the GHZ experiment, first introduced in the paper *Bell’s theorem without inequalities* (free version). I will be presenting it with as little math as possible, and essentially none of the quantum mechanical formalism. All we will consider are the outcomes of the experiment, and that will be enough. Let us begin.

## Imagine the following experiment:

You, me and Einstein, are positioned in three different buildings located far away from each other. Inside each building is a device with a screen and two buttons, "Left" and "Right". We are each tasked with choosing a button, pushing it, and recording both the outcome that appears on the screen and which button we chose to push. We quickly observe that any time a button is pushed, either an outcome of + or - appears on the screen. We set to work and perform millions of these tests. Afterwards we get together for coffee and collect all our data together. We quickly make two observations about our data:

**Observation (I): **Every time two of us chose to push the left button, and the other one chose to push the right button, the product of the outcomes was always +, for example, a selection of our data may look like this:

Test number | Person 1 | Person 2 | Person 3 | Product |
---|---|---|---|---|

301 | L,- | L,+ | R,- | (-)(+)(-)= + |

4517 | R,+ | L,- | L,- | (+)(-)(-)= + |

21034 | L,+ | R,+ | L,+ | (+)(+)(+)= + |

**Observation (II):** Every time all three of us chose to push the right button, the product of the outcomes was always -, for example:

Test number | Person 1 | Person 2 | Person 3 | Product |
---|---|---|---|---|

56 | R,-1 | R,+1 | R,+1 | (-)(+)(+)= - |

3009 | R,-1 | R,-1 | R,-1 | (-)(-)(-)= - |

52146 | R,+1 | R,+1 | R,-1 | (+)(+)(-)= - |

If we collect all the instances where one person chose Left and the other two chose Right, we can construct a table of all the possible combinations of possible outcomes (with some duplicates for the sake of clarity)

Choice | L | R | R |
---|---|---|---|

1 | + | + | + |

2 | + | - | + |

3 | + | + | - |

4 | + | - | - |

5 | - | + | + |

6 | - | - | + |

7 | - | + | - |

8 | - | - | - |

Starting from this point we can reason about what would have happened if people had done things differently. In order to proceed, we must make three assumptions:

- We assume that any of the people could have chosen a different button to press, they are free agents after all, each person's choice of button is made independently of any other person's choice or any other person's outcome for that matter.
- We assume that if someone had chosen a different button, it would not have affected the outcomes of the other people's button presses, the buildings are reasonably far apart.
- Finally, we assume that there is in principle an answer to the question "What outcome would have occurred if someone had pressed a different button?" Or, to put it in another way, we assume that if a person had chosen a different button, that something should have happened.

So, what would have happened if someone had pressed a different button? Take for instance row 2 of our table.

L | R | R |
---|---|---|

+ | - | + |

If the person in the last column had chosen to press Left instead of Right, what outcome would they have got? We know by observation (I) that if two people choose Left and one person chooses Right, that the product of the outcomes should be positive. So this implies that if the person in the last column had chosen to press Left instead, they could have only got a (-):

L | R | (L)R |
---|---|---|

+ | - | (-)+ |

Here I am putting brackets around the outcomes that would have occurred if a different choice had been made.

We can continue this reasoning for all the rows and for both columns 2 and 3, applying observation (I) in every case. I encourage the reader to try this themselves before continuing, it's a bit like a Sudoku. This is what you should get:

Choice | L | (L)R | (L)R |
---|---|---|---|

1 | + | (+)+ | (+)+ |

2 | + | (+)- | (-)+ |

3 | + | (-)+ | (+)- |

4 | + | (-)- | (-)- |

5 | - | (-)+ | (-)+ |

6 | - | (-)- | (+)+ |

7 | - | (+)+ | (-)- |

8 | - | (+)- | (+)- |

Finally, we can reason about what would have happened if the person in the first column had chosen to press Right instead of Left using observation (I) as well, because we know now what the outcomes would have had to have been if the people in columns 2 and 3 has chosen to press Left. This completes our table:

Choice | L | (L)R | (L)R |
---|---|---|---|

1 | +(+) | (+)+ | (+)+ |

2 | +(-) | (+)- | (-)+ |

3 | +(-) | (-)+ | (+)- |

4 | +(+) | (-)- | (-)- |

5 | -(+) | (-)+ | (-)+ |

6 | -(-) | (-)- | (+)+ |

7 | -(-) | (+)+ | (-)- |

8 | -(+) | (+)- | (+)- |

Now we must ask ourselves. Is this conclusion compatible with observation (II)? Recall that observation (II) was that if all three people pressed the Right button, then the product of their outcomes would be -. But if we examine our table we note that the opposite is true in every scenario! So if our three assumptions are true, and observation (I) always holds, then we should sometimes see violations of observation (II). But we never do! Both observation (I) and observation (II) always hold. So something must be wrong with our assumptions.

One of the assumptions we made was that if someone had chosen a different button, it would not have affected the outcomes of the other people's button presses. But what if this were not the case? If the devices were able to coordinate themselves by communicating with one another, they could conspire to produce the results we have observed. Maybe there are signals being sent between the different buildings?

So we set up the devices so far away from each other, and we perform the tests sufficiently quickly, that the devices would have to send signals faster than light in order to execute their conspiracy. Nevertheless, we find the same results. This is what Einstein meant by spooky action at a distance.

The critical reader might balk at this and say "Well, this is just a made up experiment, isn't it possible such results do not in fact occur?" I would like to assure you now that not only is this a scenario predicted by the theory of quantum mechanics, but precisely this kind of experiment has been performed here at the IQC with the same results (http://www.nature.com/nphoton/journal/v8/n4/full/nphoton.2014.50.html, https://uwaterloo.ca/news/news/experiment-opens-door-multi-party-quantum-communication). This is how quantum physics behaves!

What kind of explanation can we provide for this kind of behaviour? It turns out it will have to be a very radical one, and it is going to have to do away with at least one of our three assumptions.

The three assumptions that lead to the above contradiction are the following:

- Each person's choice of button is completely unrelated to any other person's choice. Often this assumption is called free will, but I think that is a bit of a distracting misnomer. Really it is the absence of superdeterminism. I think a good name for it is emergent independence. The idea is that although on a fundamental level the universe is deterministic, on some level the behaviour of systems can effectively be considered independent of one another.
- No signals can travel faster than light. This assumption is called locality. The idea is that two things separated far apart in space should not be able to influence one another's behaviour without some kind of mediating system, like light or sound. There are very good reasons to believe that this is true. One good reason is that if this weren't the case the notion of causality would break down. One more good reason is that none of us are concerned about the outcome of the ongoing galactic presidential election in the Andromeda galaxy. Rest safe in the knowledge that no one is going to blame you if you vote for Kang.
- If the person had chosen a different button, then something should have happened. This assumption is usually called counterfactual definiteness, or sometimes called realism. The idea here is that it is meaningful to ask "what if?" questions. We use this kind of assumption all the time in our everyday lives, often in the context of learning from previous actions: "If we had all voted for Kodos we probably wouldn't be toiling in the neutronium mines right now." But more importantly, we use this assumption when we think about the outcomes of things we do not observe. To paraphrase an old riddle: "If someone had been around to hear the tree fall in the forest, what would it have sounded like?" Counterfactual definiteness is probably the most subtle of the three assumptions.

The physics community is very much divided on which of these assumptions are correct and which ones are not, and they have been for a long time. For the sake of full disclosure, I will tell you that I am of the opinion that the problem lies with assumption 3, but in a subtle way which I hope to explain in a later blog post. What assumption would you choose to not have in your world view? Do you have any ideas about how to make sense of the world when one of these assumptions is not true? Please let us know in the comments.