Polarization entanglement - multiple tests

[Erik Ayer]

Imagine you have a single beam of polarization-entangled photon pairs where photons of each pair have opposite polarization. This beam goes to a polarizing beam splitter. Of each pair, the vertical photon passes directly through and the horizontal photon is reflected 90 degrees left. After the beam splitter are to polarizing filters oriented at 40 degrees and anti-45 degrees relative to the axis of the beam splitter. Will the two photons of each pair either both pass the filters or not?

This question boils down to whether the entanglement is broken by the beam splitter. I don't have a good reason why it should be broken.

Now, imagine that after the BS on the vertical beam in a polarizer oriented vertically. All the photons reaching it should pass through. However, if on then place a polarizing filter oriented at 45 degrees in the horizontal beam, the polarization of those photons will be defined on the 45-degree axis. If photon pairs are still entangled, this would alter how many pass through the vertical filter.

I conclude that I am stupid. Where is the flaw in my reasoning? It seems most likely that the beam splitter breaks the entanglement, but how does it do that?

 

[Strilanc]

Imagine you have a single beam of polarization-entangled photon pairs where photons of each pair have opposite polarization.

I don't think you can make a beam of photons like that. If you could, then the polarizing beam splitter would be performing a non-reversible operation. It would send both the state 'un-entangled H photon and un-entangled V photon' and the state 'entangled HV - VH photons' to the output state 'H photon going left, V photon going up'.

You can have two separate beams with photon pairs between them being entangled, though.

Anyways, the output of the situation you defined is "There's a V photon and, separately, an H photon. They are not entangled and they are heading towards some polarizers.".

place a polarizing filter oriented at 45 degrees in the horizontal beam, the polarization of those photons will be defined on the 45-degree axis. If photon pairs are still entangled, this would alter how many pass through the vertical filter.

No, that's not how it would work. Half of the photons that pass through the V polarizer would be associated with the photons that made it through the D polarizer, but there's another half of the V-passing photons that would be associated with the photons that didn't make it through the D polarizer. The local total stays the same as when you do nothing; only the association changes.

 

[Erik Ayer]

A BBO can be cut for "co-linear" downconversion. The two cones of downconverted light intersect at a point where photons have the same wavelength and are entangled in polarization. The alternative is for the cones to overlap in two places such that you get two separate beams that are polarization entangled.

Thank you for the reply. I will need to think on this for a while.

What's really capturing my attention in the quantum eraser experiment where polarizing filters are placed before the slits of a double-slit. Photons are either absorbed or they pass through, and the ones that pass through are marked as to which way they went. Mucking with the entangled partners "erases" the which-way information - forces them to have definite polarization in the 45-degree plane, but 50-50 in the plane of the double-slit.

Could this be more simple and more definite? What if there are two separate beams with opposite polarizations, and each goes into a polarizing beam splitter. For each pair, one will be transmitted and the other reflected to the left (or right). After one beam splitter on the transmitted output could be another beam splitter, and all the photons reaching it should also be transmitted. If a 45 degree filter follows the reflected output of the other beam splitter, that should erase the polarization of both, and the light will split at the second beam splitter on the original side. This is all to illustrate a case where there are not two uses of the entanglement, for lack of a better term.

 

[Strilanc]

The two cones of downconverted light intersect at a point where photons have the same wavelength and are entangled in polarization.

But don't they have different directions? They coincide at a point, but not along a beam?

Could this be more simple and more definite? What if there are two separate beams with opposite polarizations, and each goes into a polarizing beam splitter. For each pair, one will be transmitted and the other reflected to the left (or right). After one beam splitter on the transmitted output could be another beam splitter, and all the photons reaching it should also be transmitted. If a 45 degree filter follows the reflected output of the other beam splitter, that should erase the polarization of both, and the light will split at the second beam splitter on the original side. This is all to illustrate a case where there are not two uses of the entanglement, for lack of a better term.

You should be working with the kets instead of trying to reason about it at the level of "polarizers break the entanglement".

As soon as you split the photons into vertical and horizontal groups, there's no more entanglement between the groups. You need a state like $|HV\rangle + |VH\rangle$.

 

[StevieTNZ]

Thank you for the reply. I will need to think on this for a while.

Perhaps you could consider the use of wave plates rather than polarizing filters.

 

[Erik Ayer]

Take a look at the wiki: https://en.wikipedia.org/wiki/Spontaneous_parametric_down-conversion

Most of the way down it shows two cones of downconverted light, and these are where the wavelength is twice that of the pump. The separation angle of the cones can be adjusted so they intersect along a line - the entangled photons are all in the same beam.

As for kets: physics was one major of three, and I didn't get into quantum hardly at all. Entanglement got me interested a bit more than a decade ago and I've read a lot of stuff online. However, learning the formalism has been a bit more difficult. Quantum Physics For Dummys was no help at all. Can you recommend some good sources for learning bras and kets? I would greatly appreciate it! As you indicated, I'm limited to a more qualitative view and analysis.

As for waveplates, they alter the polarization of photons rather than filtering, correct? These would maintain levels of light better than filters, and it would be interesting to see whether altering or filtering the entangled pairs would erase path information.

 

[Strilanc]

Can you recommend some good sources for learning bras and kets? I would greatly appreciate it! As you indicated, I'm limited to a more qualitative view and analysis.

You can learn a lot about the math behind quantum information by watching the Khan-style video series Quantum Computing for the Determined, or reading the textbook co-authored by the creator of that series.

 

[Erik Ayer]

Ok, I think I understand the wave plate. It alters the polarization without breaking the entanglement, and that's why it is used in front of each slit of the quantum eraser. Then, the filtering of the entangled photons at 45 degrees forces the double-slit photons to change their polarization to be defined at 45 degrees, thus erasing the "marking" done by the quarter wave plates.

So the polarizing beam splitter DOES break the entanglement, making erasure impossible.

I am clawing my way towards understanding. Thanks again!

 

[Erik Ayer]

Strilanc, thank you for the link! I'll buy the book.