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
Ax or By with {x,y}∈{1,2} where 1 and 2 represent the
two questions and {ax or by}∈{−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.
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.
- 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…
- 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.
- 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?
Amazing information on Buddhism. Thanks for sharing your post
ReplyDeleteI recently converted to Buddhism and have been researching and reading extensively about Buddhism and Antique Buddhas . I came over to your post and enjoyed it. Please keep posting.
ReplyDeleteI read all of your posts and was blown away by the quality of information you provide. You did an amazing job depicting physics and Buddhism. The connection between these two subjects now makes perfect sense to me.
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