I What happens to entanglement inside a polarizer?

[boxfullofvacuumtubes]

Suppose you prepare two polarization-entangled horizontally polarized photons.

Scenario 1:

After the first photon passes through a linear polarizer oriented at 45 degrees, it will have later a 50% probability of being measured as horizontally polarized and 50% as vertically polarized. The second photon will retain the 100% probability of being measured as horizontally polarized.

What happens to entanglement in this case?

Are the two photons still entangled, but the correlation is now between the first photon having the 45-degree polarization and the second photon having the horizontal polarization? Would such a state be described as $\frac{1}{\sqrt{2}}\left(\left|H,H\right> + \left|V,H\right>\right)$?

Or, did the 45-degree polarizer break the initial entanglement by measurement of the first photon's polarization?

Scenario 2:

Suppose the first photon does not pass through a linear polarizer oriented at 45 degrees, but is absorbed by the polarizer instead (which can happen with a 50% chance for LHP). What happens to entanglement? Was the entanglement between the two photons simply destroyed as the first photons was absorbed? Or, is the second photon now entangled with the polarizer? Or... something else?

 

[DrChinese]

1. Suppose you prepare two polarization-entangled horizontally polarized photons.

2. Suppose the first photon does not pass through a linear polarizer oriented at 45 degrees, but is absorbed by the polarizer instead (which can happen with a 50% chance for LHP). What happens to entanglement? Was the entanglement between the two photons simply destroyed as the first photons was absorbed? Or, is the second photon now entangled with the polarizer? Or... something else?

1. If they are horizontally polarized, they cannot also be polarization entangled. That is a contradiction.

2. Precisely when (and how) entanglement is broken is unknown. Although entanglement might be presumed to end upon irreversible measurement of the first photon, there is nothing obvious that occurs to the second that can be observed in isolation. When certain correlations between the two are calculated, then the entangled state statistics will be seen.

 

[boxfullofvacuumtubes]

1. If they are horizontally polarized, they cannot also be polarization entangled.

My bad. It should have read: Suppose the polarization-entangled photons are initially in a state $\frac{1}{\sqrt{2}}\left(\left|H,H\right> + \left|V,V\right>\right)$, and the first photon passes through a linear polarizer oriented at 45 degrees. Did the 45-degree polarizer break the initial entanglement by measuring the first photon's polarization? If we measure the H/V polarization of both photons afterwards, there is no correlation, right?

What I'm trying to wrap my head around in the case of scenarios #1 and #2 is: We can no longer detect entanglement of the two photons by measuring their H/V polarizations. But is the mechanism the same? Do both passing through a 45-degree polarizer and being absorbed by a 45-degree polarizer count as a measurement of polarization? Is there anything fundamentally different about how entanglement becomes broken in these cases?

 

[DrChinese]

My bad. It should have read: Suppose the polarization-entangled photons are initially in a state $\frac{1}{\sqrt{2}}\left(\left|H,H\right> + \left|V,V\right>\right)$, and the first photon passes through a linear polarizer oriented at 45 degrees. Did the 45-degree polarizer break the initial entanglement by measuring the first photon's polarization? If we measure the H/V polarization of both photons afterwards, there is no correlation, right?

What I'm trying to wrap my head around in the case of scenarios #1 and #2 is: We can no longer detect entanglement of the two photons by measuring their H/V polarizations. But is the mechanism the same? Do both passing through a 45-degree polarizer and being absorbed by a 45-degree polarizer count as a measurement of polarization? Is there anything fundamentally different about how entanglement becomes broken in these cases?

Going through a polarizing beam splitter (PBS) would also break the entanglement. Anything that would serve to provide polarization information, in principle, would also do it.

Let's go back to your example. Once the first photon is determined to be polarized at 45 degrees, it is a 100% certainty that the other one is likewise polarized at 45 degrees.

 

[Nugatory]

My bad. It should have read: Suppose the polarization-entangled photons are initially in a state $\frac{1}{\sqrt{2}}\left(\left|H,H\right> + \left|V,V\right>\right)$, and the first photon passes through a linear polarizer oriented at 45 degrees. Did the 45-degree polarizer break the initial entanglement by measuring the first photon's polarization?

You've written the state in the H/V basis, which is convenient if this two-photon system is going to interact with a vertically or horizontally oriented polarizer, but not so convenient when it's going to interact with a polarizer at a 45-degree angle. If you rewrite the state using the +45/-45 basis it will be clear what comes out of an interaction with the 45-degree polarizer.

What I'm trying to wrap my head around in the case of scenarios #1 and #2 is: We can no longer detect entanglement of the two photons by measuring their H/V polarizations. But is the mechanism the same? Do both passing through a 45-degree polarizer and being absorbed by a 45-degree polarizer count as a measurement of polarization? Is there anything fundamentally different about how entanglement becomes broken in these cases?

Yes, yes, no. You started with a quantum system in the state $\frac{1}{\sqrt{2}}\left(\left|H,H\right> + \left|V,V\right>\right)$. It interacted with a polarizer, and however that interaction turns out, the wave function will collapse to a state that is factorizable, not entangled (but don't take my word for this! Write down the post-collapse wave function for both cases, see for yourself).