Bell tests: Changing detector settings "just before photon arrives"

[greypilgrim]

Hi.

To close the locality loophole in Bell tests, detector settings may be changed randomly "just before the photon arrives". Or I have read about a delayed choice quantum eraser with a Mach Zehnder interferometer where the second beam splitter was put into the setup or taken out "when the photon has already passed the first beam splitter".
How well-defined are such statements about the position of the photon when it isn't actually measured?

What also confuses me that the mathematical treatment of Bell tests usually only involves the time-independent part with the entangled polarizations (or spins in Stern-Gerlach setups). Is there also a time-dependent part that justifies such statements about position, and how does it look like?

 

[gentzen]

How well-defined are such statements about the position of the photon when it isn't actually measured?

Those statements are well-defined as part of measurement protocols. (As you noticed, you cannot actually know the moment "just before the photon arrives", not just because you didn't measure "the position of the photon" or "the moment it was emitted", but because the photon would arrive at the detector before the results of such measurements, or the actions they could have triggered.) Such statements can only mean that you randomly change the detector settings with a frequency (sufficiently) higher than c/d, where d is the distance between source (or beam splitter) and detector, and c is the speed of light.

Is there also a time-dependent part that justifies such statements about position, and how does it look like?

There is a time-dependent part in non-relativistic QM, but it is mostly irrelevant for the justification of such statements. It is unclear to me whether a QED/QFT treatment would give you a the time-dependent part, but I don't think that its absence would be a problem for justification.

 

[greypilgrim]

Let's assume two silver atoms get spin-entangled and sent through two Stern-Gerlach apparatuses. I guess atoms are heavy enough for an approximate non-relativistic description where the position expectation value isn't too far off from a classical trajectory (but I might very well be wrong about that).

Just how will the full wavefunction look mathematically? Is it a product of the entangled spin part and the two position wavefunctions, i.e. something like
$\ket{\Psi}=\ket{\text{Bell state}}\ket{x_1}\ket{x_2}\ ?$
Or are the positions entangled as well?

 

[gentzen]

Let's assume two silver atoms get spin-entangled and sent through two Stern-Gerlach apparatuses. I guess atoms are heavy enough for an approximate non-relativistic description where the position expectation value isn't too far off from a classical trajectory (but I might very well be wrong about that).

I agree.

Just how will the full wavefunction look mathematically? Is it a product of the entangled spin part and the two position wavefunctions, i.e. something like
$\ket{\Psi}=\ket{\text{Bell state}}\ket{x_1}\ket{x_2}\ ?$
Or are the positions entangled as well?

Basically yes. The positions will only get entangled behind the Stern-Gerlach apparatuses. For describing the actual experiment, using momenta instead of positions would be more precise, i.e. $\ket{\Psi}=\ket{\text{Bell state}}\ket{p_1}\ket{p_2}$. The momenta can be controlled more accurately than the positions, and also the Stern-Gerlach apparatuses themselves act primarity on the momenta, and only indirectly (via the momenta) on the positions.

But of course, I know that you are more interested in the positions, because you want to change the settings (i.e. their rotation) of the Stern-Gerlach apparatuses just before the solver atoms arrive. That is fine too, even so the math could get a bit complicated, because you still need to describe the momentum being more accurately determined than the position. It can be done, I am just not sure about the most straightforward way to do it.

 

[PeterDonis]

Is it a product of the entangled spin part and the two position wavefunctions

No. The configuration space states of the silver atoms are not even close to being eigenstates of position. They are pretty close to being eigenstates of momentum, but are better thought of as wave packets with a narrow spread around the classical momentum value.

The positions will only get entangled behind the Stern-Gerlach apparatuses.

An SG apparatus entangles the momentum and spin, not the position and spin.