How to determine up and down far away?

[Genady]

Let's take a pair of entangled electrons in the state |ud>-|du>. If the first electron measures up, the second is down, and vice versa. Let's keep one electron on Earth and let's send another one, very carefully, to a planet far away, say in the M87 galaxy. They are still entangled, so if the first electron is up the second is down... But what are the directions 'up' and 'down' on that planet? How do they relate to the direction of the measurement here on Earth?

[Markus Hanke]

What is correlated in this entanglement relationship is the relative orientation of the spin vectors, so it doesn’t matter how one locally defines the direction of spin measurement. The crucial point is that whatever direction you measure it in, the correlated particle will come out as opposite in terms of relative orientation. In some sense it’s only the “sign” that’s important. 

[Genady]

24 minutes ago, Markus Hanke said:

What is correlated in this entanglement relationship is the relative orientation of the spin vectors, so it doesn’t matter how one locally defines the direction of spin measurement. The crucial point is that whatever direction you measure it in, the correlated particle will come out as opposite in terms of relative orientation. In some sense it’s only the “sign” that’s important. 

Thank you, I understand this. My question is, what determines relative orientation between two localities in two different places in the universe? What determines that two directions in these two localities are parallel? Is it determined according to some kind of parallel transport? If the space were flat the answer would be obvious, but it is not.

I would guess, that in the curved and dynamic spacetime, we would need to establish parallel spatial directions between locality 1 at time t1 of one measurement and locality 2 at time t2 of the other measurement. In these two directions, the results of the measurement will be opposite.

OK, 40 minutes later, I've solved it. The relative orientation could be determined by a gyroscope. We can send the entangled electron to M87 together with a gyroscope, which is prepared in parallel to a gyroscope that stays on Earth. Then we measure each spin in the direction of the respective local gyroscope's axis.

Edited May 20, 2022 by Genady

[Markus Hanke]

23 hours ago, Genady said:

What determines that two directions in these two localities are parallel? Is it determined according to some kind of parallel transport? If the space were flat the answer would be obvious, but it is not.

I’m afraid I don’t think I understand your question. Spin as a property is relevant only on subatomic or at most atomic scales - on those scales spacetime will, for all practical purposes, be flat, unless you are in an extreme gravitational environment. Were you referring to such exceptional scenarios? If not, then you are only comparing two small patches of spacetime that are locally flat, so there should be no difficulty.

It should also be remembered that the components of the spin vector do not commute, so you can only ever know for definite one of the components at a time, plus the squared magnitude of the vector (these do commute). So when we speak of the ‘orientation’ here, we really are talking about eigenvalues of the respective spin operator, rather than a classical, well defined vector with known components. So the only difference can be a sign, since the squared magnitude is immutable. The spectrum of these operators shouldn’t depend on the metric of the background spacetime.

[Genady]

I'll try to describe my question in a more concrete setting. Let's say we have two Stern–Gerlach apparatuses, side by side, and two entangled electrons as before. The apparatuses are parallel to each other. We measure one electron in one apparatus and find out that it moves toward the magnetic N of the apparatus. If we measure the second electron in the second apparatus, it will move to its magnetic S.

If the second apparatus were set e.g. perpendicularly to the first, then the second electron could move there to the N or to the S with equal probabilities.

Now, the second apparatus is in M87 and we want it to be set parallel to the first for the purpose of the measurement of the second electron spin. How do we determine the correct setting for the second apparatus? Wouldn't it depend on the path of the electron from Earth to M87 through the curved space? 

[Markus Hanke]

Ok, so basically the question is whether or not an existing entanglement relationship is effected by changes in gravity (eg through moving one of the entangled subsystems into a different gravitational environment)?

That’s a truly excellent question, and I will admit that I don’t know the answer for sure. Based on basic principles, I would have to say yes - an entangled system such as the one described here is mathematically a sum of tensor products, and in curved spacetime such products would explicitly depend on the metric, which should change the correlations as compared to Minkowski spacetime. A quick online search seems to confirm this, and experiments have been proposed to test this effect:

https://iopscience.iop.org/article/10.1088/1367-2630/16/5/053041

Whether or not there is path-dependence will probably depend on the background metric.

[swansont]

Up and down are dictated by the local magnetic field

22 hours ago, Genady said:

Now, the second apparatus is in M87 and we want it to be set parallel to the first for the purpose of the measurement of the second electron spin. How do we determine the correct setting for the second apparatus? Wouldn't it depend on the path of the electron from Earth to M87 through the curved space? 

You have to transport the electron in such away that you don’t collapse the wave function, but beyond that it doesn’t matter. The spin is undetermined, and what matters is the field when you do the measurement.

[Genady]

12 minutes ago, swansont said:

Up and down are dictated by the local magnetic field

You have to transport the electron in such away that you don’t collapse the wave function, but beyond that it doesn’t matter. The spin is undetermined, and what matters is the field when you do the measurement.

I know how the magnetic field was set on Earth when the other entangled electron was measured 'up'. I want to set the magnetic field here (M87) so that the electron here will definitely measure 'down'. If 'here' was in the same lab, I would set the magnetic field just parallel to the other.

[swansont]

6 minutes ago, Genady said:

I know how the magnetic field was set on Earth when the other entangled electron was measured 'up'. I want to set the magnetic field here (M87) so that the electron here will definitely measure 'down'. If 'here' was in the same lab, I would set the magnetic field just parallel to the other.

Right, but you don’t have to do it that way. If you went to the lab next door, your field could be at any arbitrary angle to the other measurement. Up and down are relative to however you set up the quantization axis where you do the measurement. 

If the spin was aligned with the N pole in your lab on earth, measuring along that axis - however it is aligned with regard to anything else - will give you your down spin in the lab at M87. 

 

[bangstrom]

On 5/22/2022 at 5:57 PM, Genady said:

I know how the magnetic field was set on Earth when the other entangled electron was measured 'up'. I want to set the magnetic field here (M87) so that the electron here will definitely measure 'down'. If 'here' was in the same lab, I would set the magnetic field just parallel to the other.

The first observation establishes the orientation of both particles at the instant of observation. That is, the event is non-local. The very first observation destroys the entanglement so, if you know the orientation of the electron in your laboratory, you can’t predict the orientation at another location because the particles are no longer entangled.

 

[joigus]

Here's the problem (my emphasis):

On 5/20/2022 at 10:12 PM, Genady said:

If the first electron measures up, the second is down, and vice versa.

Now you say (my emphasis):

On 5/20/2022 at 10:12 PM, Genady said:

Let's keep one electron on Earth and let's send another one, very carefully, to a planet far away, say in the M87 galaxy.

In order to keep an electron on Earth for as long as it takes to send the other one all the way down to M87, you must make it orbit under a field. That's a lot of change in the quantum mechanical phase of the electron.

And, further (my emphasis again),

On 5/20/2022 at 10:12 PM, Genady said:

They are still entangled, so if the first electron is up the second is down...

This conflicts with previous point that the electron measures up. If they measure anything (any one of them), they're no longer entangled, and their common state is described by a density matrix representing a strict mixed state.

But the question whether the gravitational field affects the quantum description and how is outstanding, and I don't know the answer to it, but I'm thinking about a possible gedanken to make an equivalent question without the problems I see.

It's questions like this that make me keep coming back to these forums, to the detriment of my activity in other 'more expert-driven' forums.

I just would like to emphasize Swansont's particular point that,

On 5/23/2022 at 1:12 AM, swansont said:

Up and down are relative to however you set up the quantization axis where you do the measurement. 

which I think is essential in all this business.

[swansont]

2 hours ago, joigus said:

This conflicts with previous point that the electron measures up. If they measure anything (any one of them), they're no longer entangled, and their common state is described by a density matrix representing a strict mixed state.

An important point. They must still be entangled for the explanation I gave to hold. The measurement has to happen after the second electron reaches its destination.

Holding the electron has to happen in such a way that you aren’t collapsing the wave function (i.e. breaking the entanglement)