Undergrad Why Bohmian Mechanics needs non-locality?

[Deepblu]

I always think about entanglement as pure conservation of energy and conservation of angular momentum. In fact I see that only conservation of energy is non-local, and that quantum objects have nothing to do with non-locality, for example you can entangle 2 atoms that have never interacted with each other or have never been close to each other by collecting the photons that they emit and then bringing the photons together on a beamsplitter.

So my question is why BM need super non-locality, where each particle can affect other particles in the universe instantly, while what is going on is just a blind conservation of energy?

Is my way of thinking about non-locality as consequence of conversational laws and has nothing to do with quantum objects themselves correct?

Thanks,

 

[Demystifier]

Conservation of energy is a local law. To see why entanglement is non-local you should read about the Bell theorem.

 

[stevendaryl]

As far as I know, there are no strict conversational laws in physics. That's why we have moderators.

To illustrate why people say that it would require nonlocal interactions in order to explain quantum mechanics in a realistic fashion, the best example is the EPR experiment.

Particles have an associated intrinsic angular momentum called spin. It's a vector quantity with a direction and a magnitude. In EPR, you have some way to produce particles in pairs: one electron and one positron, such that the total spin adds up to zero.

Whenever you measure spin, you can't measure the vector directly, you can only measure components of the spin along particular axes. So you can measure, say, the x-component of the spin, the y-component, the z-component (or the component of the spin along any other direction). The weird fact about spin is that for electron or positron, you always get either +1/2 or -1/2 (in units of $\hbar$).

So if the total spin for the pair is zero, that means that if you measure the electron's spin along any direction, you always get the opposite of the spin of the positron along that same direction.

The question: How do you explain this perfect anti-correlation?

It's easy to explain using nonlocal interactions: When you measure the spin of the electron along the x-axis, if you get +1/2, then the spin of the positron along that axis flips to -1/2. If you get -1/2, then the spin of the positron flips to +1/2. That's nonlocal, because the electron and positron can be arbitrarily far away from each other when their spins are measured.

You can try to explain the correlations in a local way. Maybe the electron and positron are pre-programmed to always give the opposite result when measured on the same axis. It's as if they agree ahead of time: "Okay, for the x-axis, I'll give the result +1/2, and you give the result -1/2. For the y-axis, I'll give -1/2 and you give +1/2..." It certainly isn't obvious, but it can be proved that there is no set of pre-programmed instructions that could possibly give the predictions of quantum mechanics. Quantum mechanics besides predicting perfect anti-correlation for measurements along the same axis, predicts weaker correlations for measurements along different axes. If you measure the electron along axis $A$ and measure the positron along axis $B$, then QM predicts that you will get the same result with probability $sin^2(\theta)$ and opposite results with probability $cos^2(\theta)$ (where $\theta$ is the angle between $A$ and $B$). There is no way to get these results using pre-programmed instructions.

 

[Deepblu]

Conservation of energy is a local law. To see why entanglement is non-local you should read about the Bell theorem.

But in entanglement conservation of energy is non-local, right?

 

[Deepblu]

@stevendaryl
I have no problem with non-locality. And I 100% believe that entanglement is non-local and that no local hidden variable theory can explain it.

My problem is in defining which part of the quantum system is non-local, according to my understanding (or intuition) only conversational laws are non-local, not the quantum objects themselves (Like in the 2 atoms entanglement example I described in my first post)

 

[Demystifier]

But in entanglement conservation of energy is non-local, right?

Wrong.

 

[stevendaryl]

@stevendaryl
I have no problem with non-locality. And I 100% believe that entanglement is non-local and that no local hidden variable theory can explain it.

My problem is in defining which part of the quantum system is non-local, according to my understanding (or intuition) only conversational laws are non-local, not the quantum objects themselves (Like in the 2 atoms entanglement example I described in my first post)

I'm not sure what you mean by "parts". It's the experimental results that are nonlocal. I don't see how you can attribute it to anyone thing.

 

[Deepblu]

Wrong.

Then how it comes that sum of measurments of A,B after entanglement always show conserved quantity?

 

[Nugatory]

My problem is in defining which part of the quantum system is non-local, according to my understanding (or intuition)...

Do not trust your intuition here - it is based on a lifetime of experience with things that do not behave quantum mechanically

...only conversational laws are non-local, not the quantum objects themselves (Like in the 2 atoms entanglement example I described in my first post)

In an entangled system, you don't have two quantum systems, each describing one particle. You have a single quantum system with a single wave function. As far as the mathematical formalism is concerned, we aren't performing measurements on two quantum objects, we're performing two measurements on a single system. Non-locality is inevitable when the two measuring devices are physically separated.

 

[Demystifier]

Then how it comes that sum of measurments of A,B after entanglement always show conserved quantity?

If energy was the only measured observable, then one could argue that particles had their energies even before measurement and that the measurement only revealed those preexisting values. That would be a local explanation of energy conservation, very much like energy conservation in classical mechanics.

 

[Nugatory]

Then how it comes that sum of measurments of A,B after entanglement always show conserved quantity?

Do they?

If I generate a large number of number of spin-entangled particle pairs (yes, I know that in the post just above I said that's not what the mathematical formalism says - that just goes to show how hard it is to properly describe quantum systems using natural language - better would be "a quantum system prepared in the state $\frac{\sqrt{2}}{2}(|ab\rangle+|ba\rangle)$") and then set up my two spin measuring devices so that they are not quite parallel. When the left-hand detector reads spin-up, for some pairs the right-hand detector will read spin-up and for others it will read spin-down. How are both results consistent with something being conserved?

This isn't to say that conservation laws don't work with quantum mechanics - they do, but not necessarily in the way that your classical intuition about systems of two particles would suggest.

 

[Deepblu]

If energy was the only measured observable, then one could argue that particles had their energies even before measurement and that the measurement only revealed those preexisting values. That would be a local explanation of energy conservation, very much like energy conservation in classical mechanics.

But you know that is not true because of superposition.

 

[Nugatory]

This thread has drifted a fair amount, especially considered that the question was answered in the first reply (which, as far as I can tell, has been completely ignored by the original poster).

So let's try again...

The question is "Why Bohmian mechanics needs non-locality?"

Bohemian mechanics predicts that Bell's inequality is violated in some situations. Bell's theorem shows that any theory that predicts a violation of Bell's inequality will necessarily be non-local. Therefore Bohmian mechanics is necessarily non-local.
If one would prefer an argument that at least looks at the theory to see where non-locality enters into it, consider the form of the guiding equation of Bohmian mechanics: the state of every particle of a multi-particle system enters into it. This is discussed in some depth in one of the essays in Bell's "Speakable and unspeakable in quantum mechanics", and there's also a good description at https://plato.stanford.edu/entries/qm-bohm/

 

[atyy]

@stevendaryl
My problem is in defining which part of the quantum system is non-local, according to my understanding (or intuition) only conversational laws are non-local, not the quantum objects themselves (Like in the 2 atoms entanglement example I described in my first post)

Roughly speaking, the wave function is nonlocal (or not necessarily local) because it is in Hilbert space. You can imagine an identical Hilbert space and an identical wave function at every point in "real" space. When the wave function changes, it changes identically at every point in "real" space immediately.

 

[Deepblu]

My original question was misinterpreted, I am not questioning non locality (the title of the thread is confusing sorry for that), and I know about Bell inequality, and I know about the math behind entanglement..etc, I am questioning the idea of conservational laws to be non-local.

@Demystifier discussed in the correct direction, but I am still not sure how conservation of energy or angular momentum can only be local rather than non-local with superposition.

Ex: if we split a high energy photon of E=1, into 2 lower energy entangled photons A,B. Then take A which will be in superpisition of several states, after measuring A and let's say I found that it has E=0.3 then I will know immediately that B has E=0.7, although A value was not set before measurement, which seems to me as non-local conservation of energy in action, (which is also confusingly consistent with the math of describing the entangled system in single wave function rather than 2)

Thanks for you patience,

 

[entropy1]

Then how it comes that sum of measurments of A,B after entanglement always show conserved quantity?

I am not sure it does. For a single trial, measured along axises that differ, say, 45° in angle, one could measure +½ and the other -½, but not point in the same direction and therefore not add up to 0.

 

[atyy]

My original question was misinterpreted, I am not questioning non locality (the title of the thread is confusing sorry for that), and I know about Bell inequality, and I know about the math behind entanglement..etc, I am questioning the idea of conservational laws to be non-local.

@Demystifier discussed in the correct direction, but I am still not sure how conservation of energy or angular momentum can only be local rather than non-local with superposition.

Ex: if we split a high energy photon of E=1, into 2 lower energy entangled photons A,B. Then take A which will be in superpisition of several states, after measuring A and let's say I found that it has E=0.3 then I will know immediately that B has E=0.7, although A value was not set before measurement, which seems to me as non-local conservation of energy in action, (which is also confusingly consistent with the math of describing the entangled system in single wave function rather than 2)

Thanks for you patience,

Not all entangled states are nonlocal. Only entangled states that violate a Bell inequality can be considered to have no local hidden variable explanation.

https://journals.aps.org/pra/abstract/10.1103/PhysRevA.40.4277
https://journals.aps.org/pra/abstract/10.1103/PhysRevA.65.042302
https://arxiv.org/abs/quant-ph/0512088

 

[stevendaryl]

I am not sure it does. For a single trial, measured along axises that differ, say, 45° in angle, one could measure +½ and the other -½, but not point in the same direction and therefore not add up to 0.

Something pointed out in a recent Insights article by @RUTA, though, is that in the case of unaligned measurement axes, there is still a kind of conservation that happens on the average.

Suppose you have a source of entangled spin-1/2 pairs such that the total spin is 0. Pick two axes $\vec{A}$ and $\vec{B}$. You measure the first particle's spin along axis $\vec{A}$ and measure the second particle's spin along axis $\vec{B}$. Then the prediction of quantum mechanics is:

If you look at the subset of pairs for which the spin of the first particle is +1/2 in the $\vec{A}$ direction, then the average total spin in the $\vec{B}$ direction is zero.

Here's how that works out:

1. For these events, the first particle has spin $\frac{1}{2} \vec{A}$.
2. The projection of that spin on the $\vec{B}$ axis is $\frac{1}{2} cos(\theta)$ where $\theta$ is the angle between $\vec{A}$ and $\vec{B}$.
3. The second particle has (according to QM) probability $sin^2(\frac{theta}{2})$ of being spin-up in the $\vec{B}$ direction.
4. The second particle has probability $cos^2(\frac{\theta}{2})$ of being spin-down in the $\vec{B}$ direction.
5. So the average spin of the second particle along the $\vec{B}$ axis is $\frac{+1}{2} sin^2(\frac{\theta}{2}) + \frac{-1}{2} cos^2(\frac{\theta}{2}) = \frac{-1}{2} cos(\theta)$
6. The average total spin (both particles) in the $\vec{B}$ direction is then $\frac{1}{2} cos(\theta)$ for the first particle, and $\frac{-1}{2} cos(\theta)$ for the second particle, for a total of $zero$.

 

[Demystifier]

Then take A which will be in superpisition of several states, after measuring A and let's say I found that it has E=0.3 then I will know immediately that B has E=0.7, although A value was not set before measurement,

I don't think that here you can know that A value was not set before measurement. Sure, the wave function was in superposition of different energies, but it doesn't necessarily mean that the actual energy was in superposition too. Perhaps systems have additional properties which are not given by their wave function.

On the other hand, if you assume that the wave function is the only property that the system possesses, then you are right. Under this assumption, conservation of energy is sufficient to prove nonlocality.