Comparison between quantum entanglement and a classical version

[Adel Makram]

No "theory" is being used to calculate measurement probabilities. It is all about relative frequencies in many repetitions of the same experiment.

But in order to theoretically explain the correlation between different outcomes, a theorem must exist about how the measurement probability is calculated.

post #16

Later, when Bob measures the spin of his particle at direction \beta, he gets +1 with probability cos^2(\frac{\beta - \phi}{2}) and -1 with probability sin^2(\frac{\beta - \phi}{2}).

QT allows Bob to measure his particle in (+) direction with with probability cos^2(\frac{\beta - \phi}{2}). But I don't know how Bob would measure it according to the classical hidden variables theory which is for me a decisive point in understanding Bell`s theorem.

 

[Nugatory]

QT allows Bob to measure his particle in (+) direction with with probabilit). But I don't know how Bob would measure it according to the classical hidden variables theory which is for me a decisive point in understanding Bell`s theorem.

Alice sends her particle through her Stern-Gerlach device, Bob sends his particle through his Stern-Gerlach device, and they each write down the direction in which the particle was deflected. That's the measurement part, and it's not done according to any theory - we're just gathering data, to see if it matches the predictions made by the various theories. After they've done a large number of pairs, they get together and compare notes, see whether the correlations in their two lists of measurements violate Bell's inequality.
(This is also the first time that the $\sin^2\frac{\alpha-\beta}{2}$ rule will appear - they didn't need it to make their measurements).

If the inequality is violated, then the experiment we've just done tells us that we can reject any local realistic hidden variable theorem, because Bell's theorem shows that no local realistic hidden variable theory can produce results that violate the inequality.

 

[Adel Makram]

I mean how to derive the dependence of correlations ( quantum and classical) on the angle between the detectors.

 

[gill1109]

I mean how to derive the dependence of correlations ( quantum and classical) on the angle between the detectors.

According to classical theory, many different correlation functions are possible. According to quantum theory, many *more* correlation functions are possible. Much more. Quantum theory is richer than classical theory.

I wrote a short paper exploring what are the possible correlation functions according to classical theory: http://arxiv.org/abs/1312.6403

 

[Mentz114]

I mean how to derive the dependence of correlations ( quantum and classical) on the angle between the detectors.

See Chapter 6 in this book

http://www.e-booksdirectory.com/details.php?ebook=6153

 

[Heinera]

The quantum correlations are derived in section II in this paper:

http://www.phy.bme.hu/~palotas/Harrison-Bell-Inequalities.pdf

 

[stevendaryl]

There is an unstated principle in Bell's derivation, which is known as Reichenbach's Common Cause Principle. Basically, this says that if two random variables are correlated, then there must be a common cause for each. Mathematically, if you have two contingent events A and B, and

P(A \& B) \neq P(A) \cdot P(B)

then there must be some cause C in the common causal past of A and B such that

P(A \& B | C) = P(A | C) \cdot P(B | C)

In other words, if you knew all the facts in the past relevant to A and B, then the probabilities would factor. This gives rise to Bell's assumption about hidden variables:

P(A \& B) = \int P(\lambda)\ d\lambda\ P(A, \lambda) \cdot P(B, \lambda)

My understanding is that this principle isn't logically necessary, although it is tacitly assumed in almost all reasoning about cause and effect.

 

[Nugatory]

I mean how to derive the dependence of correlations ( quantum and classical) on the angle between the detectors.

You'll find the calculation of the correlation predicted by quantum mechanics in many texts (In fact, it looks like Mentz114 and Heinera posted examples while I was composing this!).

You won't find a calculation of the correlation predicted by "the classical theory" anywhere, because there is no single "the classical theory" in this conversation. What you will find, in Bell's paper, is the proof that any theory that does not allow Bob's result to depend on Alice's choice of direction must predict a correlation that obeys Bell's inequality. The term "classical theory" means any such theory; we don't need any specific example to be able to follow Bell's argument that all such theories must obey the inequality. (This is analogous to the way that I can prove that all right triangles obey the Pythagorean theorem without having to talk about any specific right triangle - it may help me understand if I have some specific examples of right triangles that I can look at to see the Pythagorean theorem in action, but it's not necessary for the proof).

 

[Nugatory]

There is an unstated principle in Bell's derivation, which is known as Reichenbach's Common Cause Principle.

Although it is not explicitly stated in his original derivation, it is also not exactly cunningly and subtly concealed . He does cover this issue in his later writings.

The history is somewhat relevant here. When Bell first published, the question on the table was whether QM was incomplete in the sense that the EPR authors meant. Thus, Bell's starting point was informally "Let's assume properties that would have satisfied the EPR authors...", and it's clear that the assumptions in his original paper met that requirement. Only after Bell's somewhat shocking result (and the experimental confirmation of inequality violations) was digested did people start putting serious energy into identifying precisely what those assumptions were. At some point along the way, the conversation shifted from the initial conclusion ("Sorry EPR - we know what you want and why you want it, but it doesn't exist") to a more rigorous specification of exactly which classes of theories are precluded by Bell's theorem and the experimental discovery of inequality violations.

 

[mikeyork]

The difference between classical and quantum entanglement is simply that in the quantum case the state of either top is unknown until it is measured whereas in the classical case it is known all along.

 

[houlahound]

I am really going to try and get this thread, but I still don't get how if you know there are a blue ball and a red ball in a box and you blindly remove the red one then with certainty you know the other one is blue once you know you have red in your hand.

How is this different to spin up spin down coupled particles a galaxy distance apart?

 

[DennisN]

I am really going to try and get this thread, but I still don't get how if you know there are a blue ball and a red ball in a box and you blindly remove the red one then with certainty you know the other one is blue once you know you have red in your hand.

How is this different to spin up spin down coupled particles a galaxy distance apart?

The difference is that in the quantum case (if we think of the balls as being quantum particles - please note that this is an analogy):

1. None of the two balls seems to have a definitive color before it is measured, instead they both have a "mix" of colors, and
2. The balls can have any color "in between" blue and red, and
3. We can not predict which ball will turn up as mostly "blue" and which ball will turn up as mostly "red", so to speak.

 

[wle]

I am really going to try and get this thread, but I still don't get how if you know there are a blue ball and a red ball in a box and you blindly remove the red one then with certainty you know the other one is blue once you know you have red in your hand.

How is this different to spin up spin down coupled particles a galaxy distance apart?

There's a fairly simple example game that shows the difference. Imagine instead that I give you and a friend (Bob) each a box with two drawers (drawers 1 and 2) that you can open. You and Bob can only open one of the drawers (when you open a drawer, anything in the other drawer is burned) and when you do you find either a red ball or a blue ball. You believe (or check, by playing this game many times) the following things:

• You (and Bob) will get either a red ball or a blue ball with 50% probability.
• If you open drawer 1 on your box and Bob opens drawer 1 on his box, you both find the same colour ball (i.e., you and Bob both get red or or both get blue).
• If you open drawer 1 and Bob opens drawer 2, you both find the same colour ball.
• If you open drawer 2 and Bob opens drawer 1, you both find the same colour ball.
• If you open drawer 2 and Bob opens drawer 2, you find opposite coloured balls (i.e, you get red and Bob gets blue, or vice versa).

Problem: how do I put balls in the drawers, without knowing in advance which ones you and Bob will decide to open, in order to guarantee the results I've just described?

Quantum mechanics predicts correlations that are of the same character (though not quite as extreme) as this.

 

[Adel Makram]

Later, Bob measures "spin-up" with probability \frac{1}{2}(1 + \vec{\beta} \cdot \vec{S_B}).

Do you mean spin-down? as the probability of the spin-up is \frac{1}{2}(1 - \vec{\beta} \cdot \vec{S_B}) = $sin^2(\theta/2)$ where $\theta$ is the angle between the two detectors. Also spin up means the direction of spin is along the direction of Bob`s detector.

 

[stevendaryl]

Do you mean spin-down? as the probability of the spin-up is \frac{1}{2}(1 - \vec{\beta} \cdot \vec{S_B}) = $sin^2(\theta/2)$ where $\theta$ is the angle between the two detectors. Also spin up means the direction of spin is along the direction of Bob`s detector.

No. The way that I was defining things, \vec{\beta} is the orientation of Bob's detector, and \vec{S_B} is the spin state of Bob's particle. If \vec{\beta} = \vec{S_B}, then Bob measures spin-up with 100% probability.

In the "collapse" interpretation, if Alice measures spin-up along direction \vec{\alpha}, then \vec{S_B} "collapses" so that \vec{S_B} = -\vec{\alpha}. So in that case, \frac{1}{2}(1+\vec{\beta} \cdot \vec{S_B}) = \frac{1}{2}(1-\vec{\beta} \cdot \vec{\alpha}).

 

[morrobay]

Initially, the spin directions for Alice's and Bob's particles are completely undetermined.

1. When Alice measures the spin of her particle, she randomly gets \pm 1, with 50/50 probability of each outcome.
2. If Alice measures +1 at direction \alpha, then Bob's particle "collapses" to the state with spin direction \phi = \alpha - \pi.
3. If Alice measures -1 at direction \alpha, then Bob's particle "collapses" to the state with spin direction \phi = \alpha.
4. Later, when Bob measures the spin of his particle at direction \beta, he gets +1 with probability cos^2(\frac{\beta - \phi}{2}) and -1 with probability sin^2(\frac{\beta - \phi}{2}).

This nonlocal model (the "collapse" interpretation) explains the probabilities, but at the cost of nonlocality; Bob's particle's state changes instantaneously when Alice finishes her measurement. [edit: probabilities in 5 corrected; originally there was a missing factor of 2]
[edit 2: sign of \phi in 3 was changed]

After Bob makes his one spin measurement later at direction β what additional measurement is made that detects that his particle had collapsed to state with spin direction Φ (2. or 3. above) that is then applied in the two formulas ? If there is no additional measurement the how is Φ determined ?

 

[Nugatory]

After Bob makes his one spin measurement later at direction β what additional measurement is made that detects that his particle had collapsed to state with spin direction Φ (2. or 3. above) that is then applied in the two formulas ? If there is no additional measurement the how is Φ determined ?

$\phi$ is calculated from Alice's result, not measured. After we calculate it, we use it to further calculate the probabilities that when Bob makes his measurement he will get spin-up or spin-down. Those probabilities are what we do measure, and the measured values agree with the calculated and predicted values.

After Bob's measurement his particle is no longer in the state $\phi$ - it is either spin-up or spin-down along $\beta$, the axis he set his detector to. We know that his particle was in the state $\phi$ before the measurement from the probabilities of getting spin-up or spin-down in the measurement.

 

[morrobay]

$\phi$ is calculated from Alice's result, not measured. After we calculate it, we use it to further calculate the probabilities that when Bob makes his measurement he will get spin-up or spin-down. Those probabilities are what we do measure, and the measured values agree with the calculated and predicted values.

After Bob's measurement his particle is no longer in the state $\phi$ - it is either spin-up or spin-down along $\beta$, the axis he set his detector to. We know that his particle was in the state $\phi$ before the measurement from the probabilities of getting spin-up or spin-down in the measurement.

Are the two Φ states that Bobs particle collapses to ( from 2 and 3 in post 76 ) QM axioms? I am sure not questioning the simple : Φ = α -π
And from the second paragraph : How is the state Φ defined ? Is it defined in terms of relation to detector setting β ?
From post #75 if Alice gets +1 at detector setting α then Bobs particle collapses to 1/2 (1 - β ⋅ α ) Then does this equal Φ = α - π ?
And is Bobs particle affected twice :
(1). After Alice gets ± 1 Bobs particle collapses to state Φ
(2) After Bob makes measurement particle is no longer in state Φ

* I am not questioning the spin up spin down measurement results or the sin and cos probability formulas that are in agreement.

 

[Nugatory]

Are the two Φ states that Bobs particle collapses to ( from 2 and 3 in post 76 ) QM axioms?

It's not an axiom of quantum mechanics, but it comes from the calculation of how the initial entangled state changes when it interacts with Alice's detector. And of course that calculation is based on the laws of QM.

And from the second paragraph : How is the state Φ defined ? Is it defined in terms of relation to detector setting β ?

Pick some direction (usually straight up and down) and call it zero degrees. Then $\alpha$, $\beta$, and$\phi$ are just angles relative to that direction. It doesn't matter what you call zero degrees as long as you're consistent about it.

From post #75 if Alice gets
+1 at detector setting α then Bobs particle collapses to 1/2 (1 - β ⋅ α ) Then does this equal Φ = α - π ?

You are misunderstanding #75. After Alice's measurement, Bob's particle collapses to either $\phi=\alpha$ (Alice measured -1) or $\phi=\alpha-\pi$ (Alice measured +1).

And is Bobs particle affected twice :(1). After Alice gets ± 1 Bobs particle collapses to state Φ
(2) After Bob makes measurement particle is no longer in state Φ

Yes. The state $\phi$ is a superposition of "+1 along angle $\beta$" and "-1 along angle $\beta$" (except in the special cases $\alpha=\beta$ and $\alpha=\beta+\pi$ where Alice's and Bob's detectors are either aligned or exactly opposite) and Bob's measurement causes that superposition to collapse into one of those two states.

 

[Adel Makram]

The probability that Alice measures her particle spin-up P₁=$\frac{(1+\cos(\theta1))}{2}$ and the probability that Bob measures his particle spin-up P₂= $\frac{(1-\cos(\theta2))}{2}$ where $\theta1$ and $\theta2$ are angles of Alice and Bob detectors, respectively.Classically, if the spin direction is a random variable and the outcomes of Alice and Bob measurement are independent (because they depend only on their detectors angles which are freely chosen), then the joint probability of 2 outcomes is a product of 2 probabilities of Alice and Bob measurements. P_classical=P₁P₂= $\frac{(1+\cos(\theta1))(1-\cos(\theta2))}{4}$.On the other hand, Quantum Theory sets up a definite probability of having both spin up P₁₂= $\frac{(1-cos(\theta1-\theta2))}{4}$. The last term represents a coefficient of state lup>lup> while the probability of similar outcomes P_similar= $\frac{(1-\cos(\theta1-\theta2))}{2}$as it equals the sum of coefficients of the states lup>lup> and ldown>ldown>.Comparing the two probabilities yields the probability P_similaris larger than P_classical where the ratio P_similar/P_classical>1is plotted in a 3D graph using Wolframealpha ( see that attached picture).The conclusion is the probability of similar outcomes, or the probability of having both spins-up, according to the QT is larger than the joint probability of classical theory as the quantum probability depends on $\theta1-\theta2$ which corresponds to an interaction between Alice and Bob according o the collapse interpretation.My opinion is; that interaction is not the only way to explain what happened. It is only manifested when we assume that Alice measurement changes Bob spin state which makes Bob now measures his spin-up with a probability $\frac{(1-\cos(\theta2))}{2}$. It is only manifested when we assume that Alice has come to a definite state of her spin and she gives the ball in the Bob playground to measure his spin. But if we let both observers have two probabilities of measuring their particles, spins-up, then the outcome of their measurements depends only on the angle between their two detectors $\theta1-\theta2)$. As if there is some sensor in the space that knows the angles between 2 space-like located detectors. Therefore, no need to say that Alice particle jumps into a state with the same direction of her detector or Bob spin state changed according to what Alice measures, instead, we can say that the probabilities of both outcomes depends on the angle between their detectors. The second idea indicates a sort of space sensor (variable, metric or whatever) which does not appear in the Minkowiski 4-space equation but at the same time won`t affect the quantum prediction. So it is not a classical hidden variable.

 

[Nugatory]

As if there is some sensor in the space that knows the angles between 2 space-like located detectors.

Not only as if there is a such a sensor, but also that sensor can influence the measurement results at both detectors. Any theory that requires such a sensor is pretty much by definition non-local. That's the one of the points of the exercise: No local realistic hidden variable theory will work.

 

[Adel Makram]

Not only as if there is a such a sensor, but also that sensor can influence the measurement results at both detectors. Any theory that requires such a sensor is pretty much by definition non-local. That's the one of the points of the exercise: No local realistic hidden variable theory will work.

Then do we need to rewrite the special relativity to include an additional space metric that has a non-local effect but doesn't refute the invariance of the speed of the light?

 

[Nugatory]

Then do we need to rewrite the special relativity to include an additional space metric that has a non-local effect but doesn't refute the invariance of the speed of the light?

No. The conclusion that you can draw from these Aspect/Bell experiments is that collapse interpretations don't play well with special relativity.

The essential premises behind special relativity are that the laws of physics are the same in all inertial frames and that the speed of light is invariant. From these we can conclude that if it is logically necessary that A happened before B (for example, because B was caused by A) then A must be somewhere in B's past lightcone and the two events are timelike-separated. If we make the further reasonable assumption that departure on a journey is logically required to happen before arrival at the end of the journey, we see that the departure event must be somewhere in the past lightcone of the arrival event, and therefore that the travel speed must be less than the speed of light. Conversely, if the speed exceeded the speed of light, the departure event would be outside the past lightcone of the arrival event and some observers would find that the journey illogically ended before it started. That's the argument against faster-than-light travel.

However, Alice's and Bob's measurements are not related in this way; it is not logically necessary that one of them happen first for all observers. We've been talking as if Alice measured first, and when she gets spin-up this "causes" Bob's wave function to collapse to the spin-down state, but that's just a (non-relativistic) way of imagining what's going on. If the two measurement events are spacelike-separated, then some observers moving relative to us will look at the exact same set of facts and say that Bob measured first and got spin-down, "causing" Alice's particle to collapse into the spin-up state before she measured it. Because both descriptions are equally valid, it is clear that using the word "cause" in either of them has to be a mistake... But it's not a mistake in QM, it's a mistake that we're making as we try to imagine what's going on in terms of a wavefunction collapse that transmits a causal influence.

 

[morrobay]

It's not an axiom of quantum mechanics, but it comes from the calculation of how the initial entangled state changes when it interacts with Alice's detector. And of course that calculation is based on the laws of QM.

Pick some direction (usually straight up and down) and call it zero degrees. Then $\alpha$, $\beta$, and$\phi$ are just angles relative to that direction. It doesn't matter what you call zero degrees as long as you're consistent about it.

You are misunderstanding #75. After Alice's measurement, Bob's particle collapses to either $\phi=\alpha$ (Alice measured -1) or $\phi=\alpha-\pi$ (Alice measured +1).

Yes. The state $\phi$ is a superposition of "+1 along angle $\beta$" and "-1 along angle $\beta$" (except in the special cases $\alpha=\beta$ and $\alpha=\beta+\pi$ where Alice's and Bob's detectors are either aligned or exactly opposite) and Bob's measurement causes that superposition to collapse into one of those two states.

No. The way that I was defining things, \vec{\beta} is the orientation of Bob's detector, and \vec{S_B} is the spin state of Bob's particle. If \vec{\beta} = \vec{S_B}, then Bob measures spin-up with 100% probability.

In the "collapse" interpretation, if Alice measures spin-up along direction \vec{\alpha}, then \vec{S_B} "collapses" so that \vec{S_B} = -\vec{\alpha}. So in that case, \frac{1}{2}(1+\vec{\beta} \cdot \vec{S_B}) = \frac{1}{2}(1-\vec{\beta} \cdot \vec{\alpha}).

Thanks . That clears up phi. Regarding post #75 when Alice measures spin up along direction α then Bobs particle collapses to S_B = - α
The other definition in that case is Bobs particle collapsing to Φ = α - π . So I should have correctly asked how are these two definitions equated in a numerical example ?
α = 30^ο
β = 80^ο
Φ = - 150^ο ?
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 ?

 

[Nugatory]

Regarding post #75 when Alice measures spin up along direction α then Bobs particle collapses to S_B = - α

Bob's particle does not collapse to $-\alpha$. It collapses into either spin-up along the direction $\alpha$ (Alice found spin-down along direction $\alpha$) or spin-up along the direction $\alpha-\pi$ (Alice found spin-up along direction $\alpha$). Note that spin-up along the direction $\alpha$ is the same thing as spin-down along the direction $\alpha-\pi$ and vice versa, so all we're saying here is that Bob's particle collapses to the opposite of what Alice measured.

The other definition in that case is Bobs particle collapsing to Φ = α - π . So I should have correctly asked how are these two definitions equated in a numerical example ?
α = 30^ο
β = 80^ο
Φ = - 150^ο ?

Yes, that is correct. Of course -150 is the same angle as +210, so we could have just as easily said that $\phi=\alpha+\pi$. The important thing is that $\phi$ points in exactly the opposite direction from the spin that Alice measured. If Alice set her detector to +30 and measured spin-up, then her particle has collapsed into the state in which its spin is pointing in the 30^o direction and Bob's has collapsed into the state in which its spin is pointing in the -150 (or +210) direction. If Alice set her detector to +30 and measured spin-down, then her particle has collapsed into the state in which its spin is pointing in the -150 (or +210) direction and Bob's has collapsed into the state in which its spin is pointing in the +30 direction.

 

[Nugatory]

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 ?

That's just basic trig identities. $\sin^2x=\frac{1}{2}(1-\cos{2}x)$ so if you set $x=(\beta-\phi)/2$ and remember that $\vec{\beta}\cdot\vec{\alpha}=\cos(\beta-\alpha)$ because they're both unit vectors you'll get there.

 

[Adel Makram]

But it's not a mistake in QM, it's a mistake that we're making as we try to imagine what's going on in terms of a wavefunction collapse that transmits a causal influence.

And that what made EPR-experiment a paradox.
But if we adopt the simple interpretation that only the difference between detectors angles matters, no paradox appears. In this interpretation, faster than light speed is not required because a causal influence on Bob spin-state from Alice`s wave-function collapse is not required too. What is required is a universal "space awareness" which is a knowledge about the difference between detectors angles in two space-like events. As if there is an additional term in 4D Minkowski space equation that when switching to "quantum mode" cancels the time metric and reduced it to Galilean 3D space equation. That makes me think that EPR-experiment is a real test for special relativity not for quantum theory.

 

[Heinera]

That makes me think that EPR-experiment is a real test for special relativity not for quantum theory.

It's not even a real test for SR, because there is no way Alice and Bob can use the quantum correlations for any kind of instant communication. If Bob looks at his results in isolation, he can't infer anything about Alice's setting. It is only when they compare their results and compute the correlation that they can decide that something non-classical has taken place. So, since the quantum results can't be used for signalling from Alice to Bob or vice versa, they are perfectly consistent with special relativity.

 

[Adel Makram]

It's not even a real test for SR, because there is no way Alice and Bob can use the quantum correlations for any kind of instant communication. If Bob looks at his results in isolation, he can't infer anything about Alice's setting. It is only when they compare their results and compute the correlation that they can decide that something non-classical has taken place. So, since the quantum results can't be used for signalling from Alice to Bob or vice versa, they are perfectly consistent with special relativity.

May be I meant, there must be a more general law of physics than SR and Minkowski space equation which does not only deal with light transmission but with quantum information as well.

 

[Nugatory]

And that what made EPR-experiment a paradox.
But if we adopt the simple interpretation that only the difference between detectors angles matters, no paradox appears

There is no EPR "paradox" - no apparent contradictions appear anywhere in the EPR argument. The EPR argument is that QM is incomplete, that there is an as-yet-undiscovered local, realistic, and deterministic hidden variable theory that explains why quantum mechanics works (similarly to way that the macroscopic randomness we see when we spin a roulette wheel is explained by classical mechanics and statistical methods applied to incompletely specified initial conditions). Given what was known at the time, that was a reasonable enough expectation - it took the best part of three decades for Bell to discover the impossibility of explaining QM with such a theory.

What is required is a universal "space awareness" which is a knowledge about the difference between detectors angles in two space-like events. As if there is an additional term in 4D Minkowski space equation that when switching to "quantum mode" cancels the time metric and reduced it to Galilean 3D space equation. That makes me think that EPR-experiment is a real test for special relativity not for quantum theory.

Now you're just babbling. What is this "4d Minkowski space equation"? What is a "time metric"? What is "quantum mode"?