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I'm having some issues to understand some concepts related with the polarization of photons, especifically the 'pratical' diference between a superposition of two polarizations (let's say $|H\rangle + |V\rangle$) and a single polarization ($|H\rangle$ or $|V\rangle$). I've read a topic similiar (Can we determine whether a photon's polarization is fixed or in a superposition), but this is not exactly what I'm trying to understand. My question: it's possible to create a experimental arrangement capable of making a diference between a superposition state or a single polarized state? Would it be diferent if I had a source who provides me with both $|H\rangle$ and $|V\rangle$ in a known rate? That is: can we use a arrangement of polarizers, mirrors, beam spliters and detectores to find out wether a source is providing these states? My original idea is to use two ortogonal polarizers in 45 degrees but I'm having some issues to understand the principles behind the situation.

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  • $\begingroup$ Are you asking if there's a way we can tell the difference between $|H\rangle$ and $|H\rangle+|V\rangle$? Can you explain a bit more why your question is not the same as the question in your link? $\endgroup$
    – enumaris
    Oct 18, 2018 at 17:40
  • $\begingroup$ @enumaris, exactly! It seems to me that my question is not the same as the question in the link I've posted because I want to propose a arrangement to distinguish these diferent states(my original idea is to use two ortogonal polarizers in 45 degrees ) but I'm having some issues to understand the principles behind the situation. $\endgroup$ Oct 18, 2018 at 18:15
  • $\begingroup$ You can use polarizers en.wikipedia.org/wiki/Polarizer $\endgroup$
    – Avantgarde
    Oct 18, 2018 at 18:17
  • $\begingroup$ @Avantgarde this is my original idea but I having some troubles to see the big picture $\endgroup$ Oct 18, 2018 at 18:24

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The basic idea is you will have to have multiple photons prepared identically, and make multiple polarization measurements (using a polarization filter) to be able to distinguish between $|H\rangle$ and $|H\rangle+|V\rangle$. One way I can do it would be to get a polarization filter and only allow vertically polarized light through. So I'm looking for the presence of a $|V\rangle$ state. If I make 10,000 measurements and I find no component $|V\rangle$ (all my results come back negative, i.e. no photons were let through the polarizer) then I am relatively certain the state was $|H\rangle$ to begin with and not $|H\rangle+|V\rangle$. If the original state was the latter, I would expect 50% of my measurements to come back positive. (If the original state was $|V\rangle$ then all 10,000 of my results should have come back positive). By building out these probabilities, I can figure out what state the photon was in originally.

This answers the question of "the photon was known to be in some pure state, how can I determine what that pure state was?". Interestingly though, when one does measure the vertical polarization, for photons which do pass through the polarizer, their state is now changed to be exactly $|V\rangle$. So say a photon starts in state $|H\rangle+|V\rangle$ and you measure $V$, then for photons that pass through your polarizer, their state is now $|V\rangle$ and NOT $|H\rangle+|V\rangle$. By measuring the state, you have changed the state (see the joke in Futurama about a "quantum finish" in a horse race). So, if one wants a bunch of $|V\rangle$ state photons, one can merely pass a bunch of randomly prepared photons through an (idealized) vertical polarizer and they will end up in that state. If one wants a bunch of $|H\rangle+|V\rangle$ photons, one can pass it through a (45$^\circ$) diagonally polarized filter to accomplish that.

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  • $\begingroup$ the thing was so much easier than I thought!!! Thank you so much!!! $\endgroup$ Oct 18, 2018 at 21:07
  • $\begingroup$ I wonder how much diferent would be the experimental arrangement if I had a scenario where the source provides me with both |H⟩ and |V⟩(but not the two at the same time) in a known rate $\endgroup$ Oct 18, 2018 at 21:28
  • $\begingroup$ @CaueEvangelista then as far as I can tell your question becomes identical to the one you linked and the answers there should provide you the answer. $\endgroup$
    – enumaris
    Oct 18, 2018 at 21:33
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    $\begingroup$ If you measure 50%, how do you know the incoming photons were all $|H\rangle+|V\rangle$, but not an equal mix of $|H\rangle$ and $|V\rangle$ from two different sources? $\endgroup$
    – safesphere
    Oct 18, 2018 at 22:41
  • $\begingroup$ @safesphere right, so in my answer I explicitly did not address the issue of there being possibly different sources. As you can see, I answered starting with "multiple photons prepared identically". If you want to distringuish between $|H\rangle+|V\rangle$ and an inhomogenous mix of $|H\rangle$ and $|V\rangle$ (essentially pure state vs non pure state) then you would have to look at intereference effects which are talked about in the OP's linked answer. I didn't want to reproduce that discussion here. $\endgroup$
    – enumaris
    Oct 18, 2018 at 22:44

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