Quantum interpretations and Buddhism Part 8: Experiment part 3 Bell's inequality

 

Bell's inequality is one of the significant milestones in the investigation of interpretations of quantum physics. Einstein didn't like many features of quantum physics, particularly the suggestion that there is no underlying physical value of an object before we measure it. Let's use Stern's Gerlach's experiment. The spin in x and z-axis are called non-commutative, and complementary. That is the spin of the silver atom cannot simultaneously have a fixed value for both x and z-axis. If you measure its value in the x-axis, it goes up, measure it in z, it forgot that it was supposed to go up in x, so if you measure in x again, you might get down. This should be clear from the previous exercise already and the rules which allow us to predict the quantum result.

 

There are other pairs of non-commutative observables, most famously position and momentum. If you measure the position of a particle very accurately, you hardly know anything about its momentum as the uncertainty in momentum grows large, and vice versa. This is unlike the classical assumption where one assumed that it's possible to measure position and momentum to unlimited accuracy simultaneously. We call the trade-off in uncertainty between these pairs as Heisenberg's uncertainty principle.

 

Niels Bohr and his gang developed the Copenhagen principle to interpret the uncertainty principle as there's no simultaneous exact value of position and momentum possible at one time. These qualities are complementary.

 

In 1935, Einstein, Podolsky and Rosen (EPR) challenged the orthodox Copenhagen interpretation. They reasoned that if it is possible to predict or measure the position and momentum of a particle at the same time, then the elements of reality exist before it was measured and they exist at the same time. Quantum physics being unable to provide the answer to their exact values at the same time is incomplete as a fundamental theory and something needs to be added (eg. hidden variables, pilot wave, many worlds?) to make the theory complete.

 

In effect, they do believe that reality should be counterfactual definite, that is we should have the ability to assume the existence of objects, and properties of objects, even when they have not been measured.

 

In the game analysis we had done, we had seen that if we relax this criterion, it's very easy to produce quantum results.

 

EPR proposed a thought experiment involving a pair of entangled particles. Say just two atoms bouncing off each other. One going left, we call it atom A, one going right, we call it atom B.

 

We measure the position of atom A, and momentum of atom B. By conservation of momentum or simple kinematics calculation, we can calculate the position of B, and momentum of A.

 

The need for such an elaborate two-particle system is because the uncertainty principle doesn't allow the simultaneous measuring of position and momentum of one particle at the same time to arbitrary precision. However, in this EPR proposal, we can measure the position of atom A to as much accuracy as we like, and momentum of B to as much accuracy as we like, so we circumvent the limits posed by the uncertainty principle.

 

EPR said that since we can know at the same time, the exact momentum of B (by measuring), and position of B (by calculation based on measurement of the position of A, clearly both momentum and position of atom B must exist and are elements of reality. Quantum physics being unable to tell us the results of momentum and position of B via the mathematical prediction calculation is therefore incomplete.

 

If the Copenhagen interpretation and uncertainty principle is right that both properties of position and momentum of a quantum system like an atom cannot exist to arbitrary precision, then something weird must happen. Somehow the measurement of the position of A at one side and momentum of B at the other side, makes the position of B to be uncertain due to the whole set up, regardless of how far atom A is from atom B. Einstein called it spooky action at a distance and his special relativity prohibits faster than light travel for information and mass, so he slams it down as unphysical, impossible, not worth considering. (A bit of spice adding to the story here.) Locality violation is not on the table to be considered.

 

Bohr didn't provide a good comeback to it. And for a long time, it was assumed that this discussion was metaphysics as seems hard to figure out the way to save uncertainty principle or locality. For indeed, say we do the experiment, we measured position of atom A first, we know the position of atom B to a very high accuracy. Quantum says the momentum of atom B is very uncertain, but we directly measured the momentum of atom B, there’s a definite value. Einstein says this value is definite, inherent property of atom B, not uncertain. Bohr would say that this is a mistaken way to interpret that exact value, momentum of atom B is uncertain, that value going more precise than the uncertainty principle allows is a meaningless, random value. Doing the experiment doesn’t seem to clarify who’s right and who’s wrong. So it’s regarded as metaphysics, not worth bothering with.

 

An analogy to spin, which you might be more familiar with now, is that two electrons are entangled with their spin would point the opposite of each other. If you measure electron A in the z-axis and get up, you know that electron B has spin down in z-axis for certain. Then the person at B measured the electron B in x-axis, she will certainly get either spin up or down in the x-axis. However, we know from previous exercise to discard the intuition of hidden variables that this means nothing. The electron B once having a value in z-axis has no definite value in x-axis, and this x-axis value is merely a reflection of a random measurement.

 

Then in 1964, came Bell's inequality which drags the EPR from metaphysics to become experimentally testable. This inequality was thought out and then experiments were tested. The violation of the inequality which is observed in experiments says something fundamental about our world. So even if there is another theory that replaces quantum later on, it also has to explain the violation of Bell's inequality. It's a fundamental aspect of our nature.

 

It is made to test one thing: quantum entanglement. In the quantum world, things do not have a definite value until it is measured (as per the conventional interpretation) when measured it has a certain probability to appear as different outcomes, and we only see one. Measuring the same thing again and again, we get the statistics to verify the case of its state. So it is intrinsically random, no hidden process to determine which values will appear for the same measurement. Einstein's view is that there is an intrinsic thing that is hidden away from us and therefore quantum physics is not complete, Bohr's view is that quantum physics is complete, so there is intrinsic randomness. Having not known how to test for hidden variables, it became an interpretation argument, not of interest to most physicist then.

 

Two particles which are entangled are such that the two particles will give correlated (or anti-correlated) results when measured using the same measurements. Yet according to Bohr, the two particles has no intrinsic agreed-upon values before the measurement, according to Einstein, they have! How to test it?

 

Let’s go back to the teacher and students in the classroom. This time, the teacher tells the student that their goal is to violate this thing called Bell’s inequality. To make it more explicit and it's really simple maths, here's the CHSH inequality, a type of Bell’s inequality:

 

The system is that we have two rooms far far away from each other, in essence, they are located in different galaxies, no communication is possible because of the speed of light limiting the information transfer between the two rooms. We label the rooms: Arahant and Bodhisattva. The students are to come out in pairs of the classroom located in the middle and go to arahant room and bodhisattva room, one student each.

The students will be asked questions called 1 or 2. They have to answer either 1 or -1. Here's the labelling. The two rooms are A and B. The two questions are A_x or B_y with {x,y}∈{1,2} where 1 and 2 represent the two questions and {a_x or b_y}∈{−1,1} as the two possible answers, -1 representing no, 1 representing yes.

 

So we have the term: a₁(b₁+b₂)+a₂(b₁−b₂)=±2. This is self-evident, please substitute in the values to verify yourself. Note: in case you still don't get the notation, a₁ denotes the answer when we ask the Arahant room student the first question a₂ for the second question, it can be -1 or 1, and so on for b...

 

Of course, in one run of asking the question, we cannot get that term, we need to ask lots of times (with particles and light, it's much faster than asking students), and average over it, so it's more of the average is bounded by this inequality. |S|= |<a₁b₁>+<a₁b₂>+<a₂b₁>−<a₂b₂>| ≤2 It's called the CHSH inequality, a type of Bell's inequality.

 

In table form, we can get possible values of say:

Questions asked	a1	a2	Separated by light years, student in B doesn’t know what student in A was asked, how student in A answered and vice versa.	Questions asked	b1	b2
A1	-1			B2		1
A2		1		B1	-1
A1	-1			B1	-1
A2		1		B2		1

 

S= |(-1)(-1)+(-1)(1)+(1)(-1)-(1)(1)|=2.

 

The goal is to have a value of S above 2. That’s the violation of Bell’s inequality.

 

Before the class sends out the two students, the class can meet up and agree upon a strategy, then each pair of students are separated by a large distance or any way we restrict them not to communicate with each other, not even mind-reading. They each give one of two answers to each question, and we ask them often (easier with particles and light). Then we take their answers, collect them and they must satisfy this CHSH inequality.

 

The students discussed and came out with the ideal table of answers:

 

Questions asked	a1	a2	b1	b2
A1, B2	1			1
A2, B1		1	1
A1, B1	1		1
A2, B2		1		-1

 

S=4, A clear violation of Bell’s inequality to the maximum.

 

So for each pair of students going out, the one going into room arahant only have to answer 1, whatever the question is. The one going to the room Bodhisattva has to answer 1, except if they got the question B₂ and if they know that the question A₂ is going to be asked of student in room arahant. The main difficulty is, how would student B know what question student A got? They are too far apart, communication is not allowed. They cannot know the exact order questions they are going to get beforehand.

 

Say if students who goes into room B decide to go for random answering if they got the question B₂, on the faint hope that enough of the answer -1 will coincide with the question A₂. We expect 50% of it will, and 50% of it will not.

 

So let’s look at the statistics.

<a₁b₁> = 1

<a₂b₁> = 1

<a₁b₂> = 0

<a₂b₂> = 0

S=2

 

<a₁b₂> and <a₂b₂> are both zero because while a always are 1, b₂ take turns to alternate between 1 and -1, so it averages out to zero. Mere allowing for randomisation and denying counterfactual definiteness no longer works to simulate quantum results when the quantum system has two parts, not just one.

 

It seems that Bell's inequality is obvious and cannot ever be violated, and it's trivial. Yet it was violated by entangled particles! We have skipped some few assumptions to arrive at the CHSH inequality, and here they are. The value for S must be less than 2 if we have 3 assumptions

 

1.There is realism, or counterfactual definiteness. The students have ready answers for each possible questions, so the random answering above is actually breaking this assumption already. These ready answers can be coordinated while they are in the classroom, for example, they synchronise their watches, and answer 1 if the minute hand is pointing to even number, and answer -1 if the minute hand is pointing to odd number. 

 

2.Parameter independence (or no signalling/locality), that is the answer to one room is independent of the question I ask the student in the other room. This is enforced by the no-communication between two parties (too far apart and so on...) Special relativity can be made to protect this assumption.

 

3.Measurement independence (or free will/ freedom) the teachers are free to choose to ask which questions and the students do not know the ordering of questions asked beforehand.

 

All three are perfectly reasonable in any classical system.

 

Violation of Bell's inequality says that either one of the 3 above must be wrong.

 

1.Most physicists say counterfactual definiteness is wrong, there is intrinsic randomness in nature or at least properties do not exist before being measured.

 

2.There are interpretations with locality wrong, deterministic in nature, but since the signalling is hidden, no time travel or faster than light that we can use. Quite problematic and challenges Special relativity, not popular but still possible based on the violation of Bell's inequality alone.

 

3.And if people vote for freedom being wrong, there is no point to science, life and the universe. Superdeterminism is a bleak interpretation.

 

Let’s go back to the game, and see if we relaxed one of the 3 rules, can the arahant and Bodhisattva room students conspire to win and violate CHSH inequality?

 

So to simulate that, say they decide to bring along their mobile phones to the questioning areas and text each other their questions and answers. Yet, this strategy breaks down if we wait until they are light years apart before questioning them, recording it, and wait for years to bring the two sides together for analysis. So for the time being, we pretend that the mobile phone is specially connected to wormholes and circumvent the speed of light no signalling limit. They easily attain their ideal scenario. S=4. We call it PR Box.

 

Actually this violation reaching to PR box is not reached by quantum particles. Quantum strangely enough only violates up to S=2.828… that means quantum non-locality is weird, but not the maximum weirdness possible. It’s this weird space of CHSH inequality violation that is non-local yet obeys no signalling. Thus the meaning of non-locality in quantum doesn’t mean faster than light signalling. We cannot use quantum entangled particles so far to send meaningful information faster than light. Quantum seems to be determined to act in a weird way, which violates our classical notion of locality, yet have a peaceful co-existence with special relativity.

 

This was a line of research which I was briefly involved in a small part during my undergraduate days. The researchers in Centre for Quantum Technologies in Singapore were searching for a physical principle to explain why quantum non-locality is limited as compared to the space of possible non-locality. So far, I do not think they have succeeded in getting a full limit, but many other insights into links between quantum and information theory arise from there and one of the interpretations involve rewriting the axioms of quantum to be a quantum information-theoretic inspired limits and derive the standard quantum physics from there.

 

The PR box example is actually the maximum non-locality that theoretical physics allows, bounded by no-signalling. So PR box still satisfy special relativity due to no signalling, however, they do not exist in the real physical world as it would violate several information-theoretic principles.

 

The PR box can be produced too if they know beforehand what questions they each are going to get, so no freedom of the questioner to ask questions. Yet, purely relaxing counterfactual definiteness cannot reproduce it. It’s because Bell’s theorem is not meant to test for purely that. We have another inequality called Leggett’s inequality to help with that (more on it later).

Puzzled by the strange behaviour of quantum, the students looked online to learn how entangled particles behave. Say using spin entangled electrons pairs, they both must have opposite spin, but whether they are spin up or down, it’s undecided until the moment they are measured. So if say electron A got measured to be spin up in z-axis, we know that electron B is spin down in z-axis immediately. With this correlation and suitable choice of angles of measuring the spin, experiments had shown that entangled particle pairs do violate Bell’s inequality, be it photon or electron. Like entangled photons (light) where we measure the polarisation angel, so the questions are actually polarisation settings which involve angles. The polarization of entangled photon pairs is correlated. A suitable choice of 3 angles across the 4 questions of A₁, A₂, B₁, B₂ allows for Bell’s inequality violation to the maximum for the quantum case. The different angles allow for more subtle distribution of probabilities to only ensure S goes to 2.828… and not more for the quantum case.

 

The students then try to simulate entangled particles without using an actual quantum entangled particle to see the inner mechanism inside it. The first idea they had was to use a rope to connect the students. Student pairs as they move to room A and B, they carry the rope along with them. When student A got question 2, student A will use Morse code to signal to student B both his answer and the question he receives, then student B can try to replicate quantum results.

 

The teachers then frown upon this method. She then spends some money from the school to actually make room A and room B to be far away. Say even send one student to Mars on the upcoming human landing on Mars mission. Now it takes several minutes for light to travel from Earth to Mars, and in that time, there’s no way for internal communication to happen between the two entangled particles. The rope idea is prevented by special relativity unless we really believe that entangled particles are like wormholes (which is one of the serious physics ideas floating out there, google ER=EPR), and that they do directly communicate with each other.

 

Quick note, even if entangled particles do internal communication, it’s hidden from us by the random results they produce in measurement. It’s due to this inherent randomness that we cannot use entanglement correlation to communicate faster than light. So any claims by anyone who only half-read some catchy popular science article title about quantum entanglement who says that with entanglement, we can communicate faster than light, you can just ask them to study quantum physics properly. Quantum non-locality is strictly within the bounds of no signalling. Don’t worry about it, it’s one of the first things undergraduate or graduates physics students try to do when first learning about it and we all failed and learnt that it is indeed due to the random outcomes of the measurement which renders entanglement as non-local yet non-signalling, a cool weird nature.

 

Experimentally, Bell’s inequality violation has been tested on entangled particles, with the distance between the two particles as far away as 18km apart, using fibre optics to send the light to another lab far far away. With super fast switching, they managed to ask the entangled photons questions far faster than it is possible for them to coordinate their answers via some secret communication. Assuming no superluminal communication between them.

 

Well, ok, no rope, so what’s so strange about correlation anyway? Classically, we have the example of the Bertlmann’s socks. John Bell wrote about his friend Dr. Bertlmann as a person who couldn’t be bothered to wear matching socks so he takes the first two he has and wear them. So on any given day, if you see the first foot he comes into the room as pink socks, you can be sure that the other sock is not pink. Nothing strange here. So what’s the difference with entanglement?

 

The main difference is, before measurement, the entangled particles can be either pink or not pink, we do not know. There’s the probabilistic part of quantum which comes in again. We call it superposition of the states of pink and not pink. For photons, it can be superposition of polarisation in the horizontal and vertical axis, for electron spin, it can be superposition of up and down spin in z-axis. Any legitimate quantum states can be superpositioned together as long as they had not been measured, and thus retain their coherence, and as long as these quantum states are commutable (can be measured together).

 

In Copenhagen picture, the entangled particles acts as one quantum system. It doesn’t matter how far away in space they are, once the measurement is one, the collapse of the wavefunction happens and then once photon in A shows a result, we know immediately the exact value of photon B. Before measurement, there was no sure answer. This happens no matter if photon A is at the distance of half a universe away from photon B.

 

This type of correlation is not found at all in the classical world. The students were not convinced. They tried to gather a pink and a red sock they have to put into a bin. Then a student blindfold himself, select the two socks from the bin, switch it around and hand them over to the student pairs who will go to room A and B, one sock each. The students put the socks into their pocket, not looking at it, and only take it out to see it and try to answer based on their correlation, if one has red, we know the other has pink immediately. The pink and red colour can be mapped to a strategy to answer 1 or -1 to specific questions. This is not the same thing as real quantum entanglement, they didn’t perform better at the game. They have counterfactual definiteness. Before asking the students what colour the socks are, we know the socks already have a predetermined colour. With predetermined answers, we cannot expect b₂ to have the ability to change answers based on different questions of A₁ or A₂. Thus no hope of producing quantum or PR box-like correlation. 

 

The teacher finally felt that the students are ready for a simple Bell’s inequality derivation. She selected three students up, each student having a label of an angle: x:0 degrees, y:135 degrees and z:45 degrees. Each student is given a coin to flip. There are only two possible results each, heads or tails. Refer to the table below for all possible coin flip results:

 

0 means tails, 1 means head. The bar above means we want the tails result. So the table shows us that we can group those with x heads and y tails (xy̅) as case 5 and 6, case 3 and 7 are part of the group of y heads and z tails (yz̅). And finally, the grouping of x heads and z tails (xz̅) are case 5 and 7. It’s obvious that the following equation is trivially true. The number of xy̅ plus the number of yz̅ is greater than or equal to the number of xz̅ cases. This is called Bell’s inequality.

 

Quantum results violate this inequality, the angles above are used in actual quantum experiments to obtain the violation. In quantum calculations, the number of measurements in xy̅ basis and yz̅ basis can be lower than the number of cases in xz̅ basis. Experiment sides with quantum.

 

To translate this to CHSH, the questions that were given to the students can have a combination of two of the three angles. So the question in room arahant can be 0 degrees (x), and the question asked in room bodhisattva can be 135 degrees:y, followed by Room A asks y, Room B ask z, Room A asks x again, Room B asks z. Notice that Room A only asks between x and y, and Room B only asks between y and z, so it fits with only two questions per room. A₁ =x,  A₂=B₁=y, B₂=z.

 

Each of run the experiment can only explore two of the three angles. The heads or tails, 0 or 1 corresponds to the student’s 1 and -1 answer. As the table shows for the coin settings, the implicit assumption is that there’s counterfactual definiteness. Even if the experiment didn’t ask about z, we assumed that there’s a ready value for them. So any hidden variable which is local and counterfactual definite cannot violate Bell’s inequality. For quantum interpretations which deny counterfactual definiteness, they have no issues with violating Bell’s inequality.

 Back to EPR, Einstein lost, Bohr won, although they both didn't know it because they died before Bell's test was put to the experiment. 

Quantum entanglement was revealed to be a real effect of nature and since then it has been utilised in at least 3 major useful experiments and technologies.

1. Quantum computers. Replacing the bits (0 or 1) in classical computer with qubits (quantum bits), which you can think of as a spin, which has continuous rotation possible for its internal state, capable of going into superposition of up and down states at the same time, and having the capability to be entangled, quantum computers can do much better than classical computers in some problems. The most famous one is factoring large numbers which is the main reason why our passwords are secure. Classical computers would take millions of years to crack such a code, but quantum computers can do it in minutes. Thus with the rise of quantum computers, we need…

 

2. Quantum cryptography. This is the encoding between two parties such that if there’s an eavesdropper, we would know by the laws of physics that the line is not secured and we can abandon our quantum key encryption. There’s some proposal to replace the classical internet with quantum internet to avoid quantum computer hacking into our accounts.
3. Quantum teleportation. This has less practical usage, but still is a marvellous show of the possibility of quantum technologies. The thing which is teleported is actually only quantum information. The sending and receiving side both have to have the materials ready and entangled beforehand. The quantum object to be teleported has to be coherent (no wavefunction collapse) to interact with the prepared entangled bunch of particles at the sending end. Then the object to be teleported is destroyed by allowing it to interact with the sending entangled particles, we do some measurements, collect some classical information about the measurement, then send it at the speed of light to the receiving end. The receiving end has only the previously entangled particles, now no longer entangled due to the other end having interacted with measurements. They wait patiently for the classical data to arrive before they can do some manipulation to transform the receiving end stuffs into the quantum information of the thing we teleported. If they randomly try to manipulate the receiving end stuffs, the process is likely to fail. The classical data sent is not the same even if we teleport the exact same thing because of quantum inherent randomness involved in the measurement process. The impractical side is that large objects like human bodies are never observed to be in quantum coherence, too much interference with the environment which causes the wavefunction to collapse. And if we want to quantum teleport a living being, it’s basically to kill it on the sending side and recover it on the receiving side. It’s not known if the mind would follow, does it count as death and rebirth in the same body but different place? Or maybe some other beings get reborn into the new body?