bell-quantum.md

Bell's inequality violation or the failure of classical determinism

Today, Alice and Bob decided to go to the Science museum to test a new public experiment proving the failure of the classical description of the world in favor of the quantum description. They have heard about quantum superposition for a long time, for example in Schrödinger's cat experiment where the cat in the box is neither (or both) dead or alive according to quantum mechanics. What a piece of shit, Bob has always thought. Theses physicists are completely crazy, such a thing is completely non sense. How could you assert, or check that the cat is in a superposition of state before one opens the box ?

But now this experiment at the museum claims to offer a chance to any pair of guys to check by themselves that the world cannot be classical.

Alice and Bob arrive at the experiment place. There are two black boxes located far from one another, with one seat in front of each one. Alice goes to the left one, and Bob to the right one. Each black box has one input (a switch) which is 1 or -1, and one output which is 1 or -1.

They agree to take part in the experiment for five minutes. The experiment starts.

Every second, they choose simultaneously a random input 1 or -1, and they write down the output given by the black box. At the end of the experiment they compare the outputs. They compute the products of their outputs for each time. Then they separate their data into 4 categories :

• The times when Alice chose the input $$1$$ and Bob chose the input $$1$$.
• The times when Alice chose the input $$1$$ and Bob chose the input $$-1$$.
• The times when Alice chose the input $$-1$$ and Bob chose the input $$-1$$.
• The times when Alice chose the input $$-1$$ and Bob chose the input $$1$$.

For each category, they compute the average of the product of their outputs. We note

• $$u_{++}$$ the computed average for the first category,
• $$u_{+-}$$ the computed average for the second category,
• $$u_{-+}$$ the computed average for the third category,
• $$u_{--}$$ the computed average for the fourth category,

Small example

This is a small example to explain the previous sentence.

Here is the outputs Alice and Bob got for each time when they both chose +1 as input.

A	B
-1	1
1	1
-1	1
-1	-1
-1	-1
1	-1

They compute the products of the output for each time

A	B	AB
-1	1	-1
1	1	1
-1	1	-1
-1	-1	-1
-1	-1	1
1	-1	-1

Now they compute the average : this gives a number between -1 and +1 (here, this gives $$u_{++}=-\frac{1}{3}$$)

End of example

Now they compute $$u_{++}+u_{+-}+u_{-+}-u_{--}$$. They find a number which is bigger than 2. A panel explains that if the world were classical they could not find a number bigger than 2.

Bob is disappointed : he expected to see a Schrödinger's cat, and this experiment is just about computing numbers. So boring! While Bob cries, Alice reads the panels explaining what is so amazing about finding a number bigger than 2.

A classical picture would predict a number less than 2

Let us model this as a random experiment. Here, an experiment is what happens every second : a choice of inputs, leading to outputs.

• The internal initial state of Alice's black box (ie the state of Alice's black box just before Alice or Bob sets the input) is a random variable $$s_a$$. For example, $$s_a$$ may be the positions and speeds of all the particles inside the black box. But it could be something completely different, maybe depending on some physics we have not yet discovered. For example, if string theory is to be true, it would be the state of the strings. So we are very general about the nature of the state.
• the internal initial state of Bob's black box is a random variable $$s_b$$
• The input chosen by Alice is a random variable $$i_a$$ which is $$1$$ or $$-1$$
• The input chosen by Bob is a random variable $$i_b$$ which is $$1$$ or $$-1$$

Here, we are not assuming that one is able to know exactly the internal states of the black boxes.

Assumptions

(HI) We assume that the choice of inputs $$i_a$$ and $$i_b$$ are independent of the initial states $$s_a$$ and $$s_b$$.

We also assume that the output of Alice's black box depends only on Alice's input choice and of its initial state. This is the classical determinism assumption. This is one of the main feature that we expect from a classical world : there should be no place for chance in physics.

More formally,

• (HA) there exists a function $$F_a$$ which, given an input i and an initial state $$s$$ of Alice's black box gives the output $$F_a(s,i)\in {-1,1}$$
• (HB) there exists a function $$F_b$$ which, given an input i and an initial state $$s$$ of Bob's black box gives the output $$F_b(s,i)\in {-1,1}$$

Notations

Some notations :

• $$a_+ = F_a(s_a,1)$$ : this is Alice's output if she would choose 1 as input.
• $$a_- = F_a(s_a,-1)$$ : the same when she would choose -1 as input.
• $$b_+ = F_b(s_b,1)$$
• $$b_- = F_b(s_b,-1)$$

All of these are random variables because $$s_a$$ and $$s_b$$ (the internal states of the black boxes) are random variables.

• $$o_a = F_a(s_a,i_a)$$
• $$o_b = F_b(s_b,i_b)$$

These random variables are the observed outputs for each experiment (recall that $$i_a$$ is Alice's random choice of input).

Less than 2, man

Now, what is it that Alice and Bob are computing after having separated the datas into 4 categories ? Recall that $$u_{++}$$, for example, is the averaged product of Alice's and Bob's black boxe outputs, where the average is restricted to the cases where Alice and Bob both chose 1 as input. More formally, $$u_{++}= E(o_a o_b | i_a = 1, i_b=1)=E(F_a(s_a,i_a) F_b(s_b,i_b)|i_a=1,i_b=1) =E(F_a(s_a,1) F_b(s_b,1))=E(a_+ b_+)$$. The middle equality holds because $$s_a$$ and $$s_b$$ are independent of $$i_a$$, $$i_b$$.

The notation $$E(X)$$ is the expectancy (the expected average) of the random variable $$X$$.

Then, $$u_{++}+u_{+-}+u_{-+}-u_{--} =E(a_+ b_+) + E( a_+ b_- ) + E(a_- b_+) - E(a_- b_-)=E(a_+ b_+ + a_+ b_- + a_- b_+ - a_- b_- )=E(a_+(b_+ + b_-) + a_-(b_+-b_-))$$

Now, we see that for each experiment, either $$b_+ = b_-$$ or $$b_+ = - b_-$$ since $$b_+, b_-\in{1,-1}$$. So either $$|b_+ + b_-| = 2$$ and $$b_+ - b_- = 0$$, either $$|b_+ - b_-| = 2$$ and $$b_+ + b_- = 0$$. Then, $$E(a_+(b_+ + b_-) + a_-(b_+-b_-)) \leq 2$$. Then, $$u_{++}+u_{+-}+u_{-+}-u_{--}\leq 2$$. This is the classical prediction.

Quantum mechanics gives a way to build the two black boxes such that $$u_{++}+u_{+-}+u_{-+}-u_{--} > 2$$ (the maximum predicted value is $$2\sqrt{2}$$). Alice and Bob just experienced that.

So at least one of the assumptions we made to model the experiment is wrong.

Locality loophole

(HA) We assumed that the function $$F_a$$ which determines the output of Alice's black box depends only on Alice's choice of input and of Alice's black box internal initial state, and not on Bob's choice of input nor Bob's black box internal state.

But it gets quite subtle here. First, Alice and Bob may not enter their inputs exactly at the same time. For example, Bob may enter his input one milisecond before Alice. Second, the blackbox may need some time (1 milisecond for example) to deliver its output once the choice of input has been made. So there is a delay of two miliseconds between the time when Bob sets his input and the time when Alice's blackbox displays its output. We could imagine that during this time, Bob's black box communicates to Alice's black box some information about its initial internal state and Bob's choice of input. Then Alice's output would also depend on Bob's choice and initial state.

This is called the locality loophole of Bell's experiments.

One way to circumvent this loophole is to position the black boxes at a large distance one from another. Indeed, the locality principle coming from relativity theory states that no signal travels faster than light. In particular, Bob's black box can't communicate its internal state or Bob's choice of input faster than light, so if Alice and Bob are separated by a distance which is bigger than the distance travelled by light during two miliseconds, the locality principle ensures that the output of Alice's black box cannot depend on Bob's choice of input or Bob's black box internal state.

So it seems that the only way to deny the truth of hypothesis (HA) is to drop the locality principle. Indeed, Bohmian mechanics is a version of quantum mechanics that has full determinism but where the locality principle is not satisfied. But for this very reason, it is not very popular among quantum scientists.

Super determinism

The super determinism thesis denies the truth of hypothesis (HI) which states that the choice of inputs $$i_a$$ and $$i_b$$ are independent of the initial internal states $$s_a$$ and $$s_b$$.

It sounds a bit crazy. It means that the random choices that Alice and Bob are not really random since they depend on the internal states $$s_a$$ and $$s_b$$ of the black boxes, they don't even know about !

There is a way to test this hypothesis though. Instead of letting Alice and Bob choose the inputs, the random choice of inputs are made by making some measurement about very far stellar objects that could not possibly depend on the internal states of the black boxes because of the locality principle

Quantum reality : classical determinism is wrong

Basic quantum mechanics denies the existence of such a function $$F_a(s,i)$$ which, given an initial internal state, a choice of input, gives without ambiguity the final output displayed by the black box. Instead, it replaces it with a random variable.

This implies that even if you know completely the state of the black box before the choice of input, even if you know which input was chosen, you could not say what would be the output. And this is not because you don't know enough of the system. It is not like when you throw a dice and argue that the result is random, because in this very case, if you know exactly how you throw the dice, you could write the equations of movement and compute on which side the dice will end.

So what is this bizarre kind of state that do not allow you to deduce with certitude what the future is going to be ? This is called a quantum superposition state. A quantum physicist would say that the black box is in a superposition of two states, one leading to the 1 output, and the other one to the -1.

To come back to the Schrödinger's cat, quantum mechanics predicts that the cat's state before one opens the box is indeterminate in the same sense : even if you knew the cat's state completely, you could not deduce whether it will be dead or alive when you will open the box.

As far as I know, there is no known Bell's like experiment that would probe the quantum superposition of a macroscopic object such as a cat. The most recent work I found about macroscopic objects was about Legget-Garg inequalities, that are inspired by Bell's inequalities.

Schrödinger's cat thought experiment shows that quantum mechanics alone fails to explain the classical behaviour that we are used to at our human scale. Note that further well-studied extension to quantum mechanics such as string theory has the same problem.

There are several interpretations of quantum mechanics that are competing among scientists that could solve this issue. As I already told in the locality loophole paragraph, Bohmian mechanics is such an (unpopular) interpretation that drops the locality principle but gets rid of indeterminism and solves the Shcrödinger's cat paradox in this way : the cat is really dead or alive before opening the box.

The many-world interpretation is the most mainstream interpretation among physicists. It is compatible with the locality principle. It interprets the Schrödinger's cat experiment as follows : when you open the box, the world is forked in two versions, one where the cat is alive, and one where the cat is dead. Judge by yourself whether determinism is true according to this interpretation. As for me, I am not convinced because I don't understand why we humans are conscious of only one world, and how consciousness chooses the next world when the world is forked Some opponents also argue that this interpretation cannot be tested because it does not predict anything more than basic quantum mechanics, so it cannot be considered as a serious scientific proposition.

Relational quantum mechanics is another interpretation among quantum physicists, initiated by Carlo Rovelli, that just says that we should accept that the course of events is relative to the observer (the measurement device) in a similar fashion that Einstein’s theory of relativity made us accept the fact that for example the speed of time is relative to the observer.

Conclusion

This experiment, described in terms of abstract black boxes, does not justify the quantum theory, but at least it shows that the classical picture is wrong (determinism + locality principle).

As for me, I am not too disturbed by dropping the determinism principle. However, the Schrödinger's cat thought experiment bothers me more : what if you replace the cat with a human ? The main unsolved problem for me is when you apply the quantum superposition to the concept of mind. For this reason, I am getting more and more interested in Bohmian mechanics that denies, if I understand it correctly, the existence of quantum superposition states.

I am still waiting for a Bell's like experiment that would involve a macroscopic object such as a Shcrödinger's cat and such that in the end you would be forced either to drop locality or to admit that the cat's state cannot be predicted before opening the box, like the experiment I described for Alice and Bob.