Polarizer that accepts entangled photons

Question

I am having a bit of trouble putting the quantum mechanics of light into physical terms.

You can send unpolarized light into a polarizer; suppose we have a photon in the state:

$| \phi_1 \rangle = x | \uparrow \rangle + y | \downarrow \rangle$

and send it through a $\downarrow$ transmitting polarizer. This would mean that the output photon (if it wasn't absorbed) would be in a state:

$| \phi_2 \rangle = | \downarrow \rangle$.

This light is now considered "polarized" to $\downarrow$

It is possible to entangle two photons, and represent them as one state like so:

$| \psi_1 \rangle = \alpha | \uparrow \uparrow \rangle + \beta | \uparrow \downarrow \rangle + \gamma | \downarrow \uparrow \rangle + \delta | \downarrow \downarrow \rangle$

Could we send this light through a $\uparrow \uparrow$ transmitting polarizer and have the polarizer only transmit entangled pairs that are in a state of $\uparrow \uparrow$?

Could we even create a light wave composed of entangled photons like this?

Answer 1

Perhaps the issue is with what it means when light is entangled. Here, since the only degree of freedom is polarization, the light is entangled in the polarization degree of freedom. That imposed constraints on what the state can look like. For example, it can be in one of the Bell states: $$ | \Psi^+ \rangle = \frac{1}{\sqrt{2}} (| \uparrow \downarrow\rangle + | \downarrow\uparrow \rangle) . $$ Physically, this state represents two photons that are correlated, when one has the $\downarrow$ polarization, the other one has the $\uparrow$ polarization, and visa versa.

The next thing is the polarizer. To perform a process whereby we only filter out one combination of polarizations, we need two polarizers, one for each photon. In this case, lets say one polarizer transmits $\downarrow$ polarization, and the other one transmits $\uparrow$, then the state of the pair of photons that pass through these two polarizers is $$ | \psi \rangle = | \uparrow \downarrow\rangle . $$ It is not entangled anymore.

Light could also be entangled in other degrees of freedom, like the spatial modes of the photons, while being uncorrelated with the polarization. If one then sends such a state through polarizers, they would not affect the entanglement. The state would remain entangled.