# Comparison between quantum entanglement and a classical version

#### Nugatory

Mentor
Yes, you will often hear the word "paradox" applied to the EPR argument. However, EPR themselves do not use that word and it doesn't even appear in the paper. Instead, they argue a syllogism: All complete theories have certain properties; QM does not have one of those properties; therefore QM is not a complete theory. It turns out that their major premise is mistaken (whatever complete theory we end up, whether QM or something else, will not have all of these properties), but this was not apparent at the time and didn't become apparent until decades later.

So in your opinion, how to explain the outcomes of Alice Bob spins-experiment and quantum correlation based on causality or based on anything?
You pick an interpretation of quantum mechanics and use whatever explanation that interpretation suggests. There's an interesting discussion of some of these issues at http://arxiv.org/abs/quant-ph/0212140.

What non-locality?

Could you explain precisely what it is about QM that you see as non-local?

There is nothing Alice can do locally that in any way affects, superluminally, the results of a distant measurement - this is true in QM too. So what is it you're seeing as being 'non-local'?
Non locality means Bob particle changed to a definite spin state when Alice particle measured instantaneously according g to the collapse interpretation.

Suppose there are two experiments set ups, one allows Bob to measure the expected outcome Of his measurment of spin direction from a singlet state and the other from an entangled state.
Knowing nothing about whether Alice, Bob measures his spin direction, up-down, and take the average. If the experiment set up is a singlet one, then Bob would have a 50% chance to have a spin up. If, instead the set up is entangled and Alice exists to the other side of the world pointing her detector to a direction antiparallel to Bob' one then Bob measures his spin up in 100% of times. Now Bob comes to know that not only the spin state s entangled but he knows the direction where Alice points her detector. They can even communicate using different angles as an agreed alphabet although no physical signal is used.

I know that saying Bob measures his spin up in 100% of times sound contradictory with what have been discussed here when we say Alice has a 50% chance to measure her spin because there should nor be any preference of Alice on Bob. But I don't know how to clear this so.

#### Simon Phoenix

Gold Member
Bob to measure the expected outcome Of his measurment of spin direction from a singlet state and the other from an entangled state.
A singlet state is just the name given to a particular kind of entangled state - usually the state that looks like ~ |01> + |10>

If, instead the set up is entangled and Alice exists to the other side of the world pointing her detector to a direction antiparallel to Bob' one then Bob measures his spin up in 100% of times
No. That's completely incorrect. Sure Bob's result will be correlated with Alice's, but Alice's result is random.

#### Simon Phoenix

Gold Member
Non locality means Bob particle changed to a definite spin state when Alice particle measured instantaneously according g to the collapse interpretation.
As others have pointed out - there's a problem with using some notion of collapse here. Is it Alice or Bob who collapses the state? The answer is that it could be either depending on the frame of reference. So there's something a bit wonky with this idea that measurement causes some state change (collapse) in this scenario.

But even so it's important to realise that the issue only occurs because of a particular interpretation we've placed on things and not really a feature of QM.

Strip away the interpretative fun and games and QM is a theory that links preparations with measurement results - that's all. It doesn't tell us what's "really" happening in between these two things in any definite sense. Some people advocate thinking of the state, or wavefunction, as merely an abstract mathematical device that possesses no physical reality. It just allows us to calculate correct measurement probabilities. When we make a measurement we gain new information and so our probabilities instantly change - nothing at all non-local or strange about that process.

The point is that we don't actually need collapse as a concept in QM and nor do we need to think of the state as being some real physical entity that evolves.

Having said that, my own personal preference is to think in terms of measurement-induced collapses of a real physical entity, because that's the easiest and most pragmatic way of thinking about QM for me. But even so, there is no real issue with non-locality since no physical observables are being changed or linked by anything FTL - in other words there are no non-local experimental consequences. Only this nebulous wavefunction or state (whatever this might be) changes instantaneously in this perspective.

#### Nugatory

Mentor
Non locality means Bob particle changed to a definite spin state when Alice particle measured instantaneously according to the collapse interpretation.
That is not what "non-locality" means.

Speaking loosely, non-locality refers to the fact that to calculate the result at either detector you need to know the settings of both detectors - any mathematical equation that correctly predicts the experimental results will include the angle between the two detectors somewhere. This non-locality does not necessarily conflict with relativity (although it will if you also assume wave function collapse, which is a good reason not to make that assumption).

Note that I did say "speaking loosely"... There are some subtleties here, and the Timson/Brown paper I linked above goes into them in more depth.

#### stevendaryl

Staff Emeritus
Also if probability of Bob measuring spin down = sin2 (β - Φ)/2)
And this formula is equated to 1/2 ( 1 - β ⋅ α ) above then can someone show this numerically also ?
It's just a trigonometric identity: $sin^2(\frac{X}{2}) = \frac{1}{2}(1 - cos(X))$

#### stevendaryl

Staff Emeritus
Ok. So how SR explains the non locality then?
Well, that's what's tricky about QM. Although some interpretations (the collapse interpretation, the Bohm interpretation) involve faster than light interactions, there is no way to use QM to send a FTL signal, and so there is no way to use QM to demonstrate a violation of SR.

#### Simon Phoenix

Gold Member
Ok. So how SR explains the non locality then?
I think it's also useful to realize that we don't actually need non-locality to see a violation of the statistical inequality we call Bell's inequality, and nor do we need entanglement. If that seems counter to everything you've read on Bell's inequality then consider the following :

In the usual scenario we have Alice <---------------- Source ----------------> Bob
Let's bring Alice closer to the source so that we have Alice <- Source ---------------->Bob

In both these cases when statistics are compared we see a violation of the BI (for appropriately chosen measurement angles)

Now let's put the source in Alice's lab. She doesn't allow any of the entangled particles to reach Bob, but she makes a measurement of one of the entangled particles as she would in the normal set-up. Now she uses the result she obtains to prepare another spin-1/2 particle and sends this off to Bob

So we have || Alice <- Source -> || ....................Bob
then Alice ---------------->Bob

So in fact we don't need the entangled source at all - Alice can just randomly prepare spin-1/2 particles in the appropriate states.

Now when we compare the statistics there's a violation of the Bell inequality but it's between Alice's state preparation and Bob's measurement.

No problems with FTL signals here - and we don't need entangled particles - yet we're still seeing a violation of a mathematical inequality (the BI)

Of course this is a different experimental set-up which doesn't carry with it the same implications for 'reality' - and it was the brilliance of EPR and subsequently Bell to focus on a simple physical system that could bring out these implications for 'reality' - but it's not the 'non-local' bit of QM (or the entanglement) that's actually giving us mathematics capable of violating the mathematical inequality. A bit hard to state precisely, sorry, but I hope you get my drift here.

#### DrChinese

Gold Member
Now let's put the source in Alice's lab. She doesn't allow any of the entangled particles to reach Bob, but she makes a measurement of one of the entangled particles as she would in the normal set-up. Now she uses the result she obtains to prepare another spin-1/2 particle and sends this off to Bob

So we have || Alice <- Source -> || ....................Bob
then Alice ---------------->Bob

So in fact we don't need the entangled source at all - Alice can just randomly prepare spin-1/2 particles in the appropriate states.

Now when we compare the statistics there's a violation of the Bell inequality but it's between Alice's state preparation and Bob's measurement.
So you are saying exactly what? Bell inequalities demonstrate that local hidden variable explanations are not viable for some experiments. Yours isn't one of them.

#### Simon Phoenix

Gold Member
So you are saying exactly what? Bell inequalities demonstrate that local hidden variable explanations are not viable for some experiments. Yours isn't one of them.
Agreed :-)

I'm sorry I was hoping that I made it clear that this different experimental set-up doesn't have implications for reality in the same way that the usual Bell set-up does. My fault.

What I was trying to get at was that in both cases the maths is identical - just that we interpret it differently in each case. For me I think it brings out that the non-commutatitivity is driving the violation in QM - and when we apply this to the usual Bell set up we also get all these interesting implications for hidden variable theories.

In a nutshell there's nothing Bob can do even when given Alice's data to distinguish whether the results are from genuine entangled particles or Alice's state preparations. This has practical implications in that we can use this in a quantum key distribution using the Bell inequality protocol, but now with single particles instead of having to use entangled sources which are practically a bit trickier.

#### Simon Phoenix

Gold Member
I think there's also a sense in which the issue of non-locality is almost a red herring. That's a little bit of an overstatement and I hope the following will make it clearer what I mean.

If we impose the condition that results 'here' do not depend on settings 'there' even if here and there are only timelike separated, then a hidden variable theory of QM of this kind would satisfy the Bell inequality. We can't rule out a hidden variable theory of physics which did allow this. So even for timelike here and there we can rule out a certain class of local hidden variable theories.

So the possible candidate hidden variable theories are those in which information about settings 'there' is transferred to 'here' and affects results 'here'. I've not seen any of these constructions which look like anything like a 'sensible' theory of physics. They are possible, but somewhat artificial (at least the ones I've seen) - even in the local case where 'here' and 'there' are timelike separated. The Bohm theory, which is non-local anyway, has some attractive features and is perhaps the most 'natural' I've seen, but even there we have to introduce this complex guiding potential.

If anyone does know of a 'natural' local hidden variable theory that works (i.e. violates the BI) in this more restricted timelike case - then I'd love to see it if you have a link.

So in my view 'classical' physics is struggling even before we hit the issue of non-locality. In this case then I think it's useful to figure out what it is about QM that's causing the violation - even in the local case which only rules out a more restricted class of local hidden variable theories.

Of course it's critical that we go that last step and consider spacelike separated 'here' and 'there' - and then we put the final nail in the coffin of local hidden variable theories.

If we impose the condition that results 'here' do not depend on settings 'there' even if here and there are only timelike separated, then a hidden variable theory of QM of this kind would satisfy the Bell inequality. We can't rule out a hidden variable theory of physics which did allow this. So even for timelike here and there we can rule out a certain class of local hidden variable theories.
So what class of local hidden variables that satisfies BI in a time-like experiment?

#### Nugatory

Mentor
So what class of local hidden variables that satisfies BI in a time-like experiment?
Any theory in which the result at either detector depends on the initial state (hidden and unhidden variables) of the particle pair and on the state (hidden and unhidden variables) of that detector at the time detection will satisfy the inequality - that IS the theorem, restated in layman's language, as the theorem derives the inequality from the premise that the theory is as I just said. The timelike or spacelike separation of the measurement events is irrelevant to this proof.

Will the quantum correlation be reproduced by putting 2 polarizers with opposite direction between the source and the detectors (one polarizer at each side of the source that pass only spin up along its direction) and repeating the experiments at different angles of both polarizers near the source and the polarizers at Alice and Bob?

Lets put 2 polarizers, one on each side of the source, Ptoward Alice ( between the source and Alice) and Ptoward Bob ( between the source and Bob). The two polarizer are in opposite direction and they only pass spin-up along their respective direction so that the spin-up( up in the +ve Z direction) from Ptoward Alice is accompanied by spin-down (-ve Z) from Ptoward Bob.
Then Alice and Bob are measuring their spins at their detectors/polarizers as in standard Bells experiment with different angles of their detectors and compare their results. Will they come to a quantum correlation?
Let`s repeat the experiment by rotation the polarizers-set at many angles from 0 to 2π ( but always keeping them at opposite direction) and compare results again. Will the result be in keeping with quantum correlation?

Clearly the BI will be violated if we only compare Ptoward Alice with the detector/polarizer at Bob or Ptoward Bob with the detector/polarizer at Alice but how about the correlation between Alice and Bob outcomes?

On one hand, the correlation between Alice and Bob would be classical in nature because the spin status has been already collapsed at the polarizer near the source and the results at Alice and Bob places will be independent as if there would be taken at two non-entanged spins at different places.
On the other hand, the orientation of spins reaching Alice and Bob will be always in opposite direction and repeating experiment at different angles of the polarizer-set at the source with no information about which angle is choozen might be similar to the fully entanglement status which should mandate results in keeping with quantum correlation.
So where did I go wrong?

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#### Nugatory

Mentor
So where did I go wrong?
Alice's measurements will depend on the angle between her detector and the filtering device on her side, but not on Bob's settings. Likewise, Bob's results will depend on the angle between his detector and the filtering device on his side, but not on Alice's settings.

(You said "polarizer" but you're talking about particle spins not polarizations, so a polarizer is the wrong device to achieve your purpose. You need something like a Stern-Gerlach device, which deflects particles with spin-up along its axis in one direction and spin-down in the opposite direction).

Alice's measurements will depend on the angle between her detector and the filtering device on her side, but not on Bob's settings. Likewise, Bob's results will depend on the angle between his detector and the filtering device on his side, but not on Alice's settings.

(You said "polarizer" but you're talking about particle spins not polarizations, so a polarizer is the wrong device to achieve your purpose. You need something like a Stern-Gerlach device, which deflects particles with spin-up along its axis in one direction and spin-down in the opposite direction).
So the correlation between Alice and Bob measurements will be classical.
But considering that both sides know nothing about the angle of the filtering device near the source relative to z-axis which is changing at random for each pair. This would simulate the mixed state of fully entangled particles, I expect the correlation to follow quantum mechanics prediction.

#### Nugatory

Mentor
This would simulate the mixed state of fully entangled particles
The probability of Alice and Bob getting the same result will not be $\sin^2\frac{\alpha-\beta}{2}$. Bob and Alice will find this out when they get together afterwards and compare the results of their measurements.

#### DrChinese

Gold Member
But considering that both sides know nothing about the angle of the filtering device near the source relative to z-axis which is changing at random for each pair. This would simulate the mixed state of fully entangled particles, I expect the correlation to follow quantum mechanics prediction.
It will not simulate an entangled state at all. The stats may match at selectively chosen angles, but that's it. One is product state stats, the other is entangled state stats.

#### mpc755

In order for there to be conservation of momentum the downconverted photon pair are created with opposite angular momentums.

Each of the pair can determine the position and momentum of the other based upon their own position and momentum.

Entanglement is each of the pair being able to determine the state of the other.

They are not physically or superluminally connected.

Their ability to determine each other's state is non-local.

#### DrChinese

Gold Member
They are not physically or superluminally connected.
Welcome to PhysicsForums, mpc755!

There are QM interpretations that include such a remote connection. The underlying physical nature of entanglement of remote particles is something of a mystery.

#### mpc755

Welcome to PhysicsForums, mpc755!
Thanks!

There are QM interpretations that include such a remote connection.
Those interpretations are incorrect.

The underlying physical nature of entanglement of remote particles is something of a mystery.
It's not a mystery. Entanglement is each of the pair being able to determine the other's state based upon their own position and momentum.

#### DrChinese

Gold Member
It's not a mystery. Entanglement is each of the pair being able to determine the other's state based upon their own position and momentum.
I guess mystery is in the eye of the beholder.

So I would ask, according to your line of reasoning: If I measure Alice's p, is entangled partner Bob's q indefinite? Because Alice's is. If I measure Alice's q, is Bob's p indefinite? Because Alice's is. If so, does the nature of a measurement on Alice change Bob? This is a bit of a mystery to myself and most.

#### mpc755

I guess mystery is in the eye of the beholder.

So I would ask, according to your line of reasoning: If I measure Alice's p, is entangled partner Bob's q indefinite? Because Alice's is.
Alice's measurement is indefinite to you. It's not indefinite to Alice or Bob.

If I measure Alice's q, is Bob's p indefinite? Because Alice's is. If so, does the nature of a measurement on Alice change Bob?
The nature of measurement on Alice does not change Bob. They are propagating with opposite angular momentums.

This is a bit of a mystery to myself and most.
Yes, because no one realized you have to understand entanglement from the point of view of Alice and Bob, not an external observer performing the measurement.

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