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I am aware that many quantum entanglement experiments are based on uncertainties related to the polarization of a single photon, which can be measured in the horizontal/vertical, clockwise/counterclockwise, or any other orthonormal basis. However, I have been reading about the nature of photons in more depth from the book The Quantum Theory of Light, by Rodney Loudon, which makes it seem that each photon is associate with a single wavevector $\mathbf k$ and frequency $\omega_{\mathbf k}$. It seems to me that the wavevector completely defines the polarization of the photon, so how can there be uncertainty as to which way the photon is polarized? I understand that the state of an electromagnetic field can be delocalized over several possible number states (i.e. different combinations of photons). If every photon is inherently associated with a single mode of the field, though, how is saying that a photon has the polarization state $\frac1{\sqrt2}\left(|\uparrow\rangle+|\rightarrow\rangle\right)$ like saying that a particle is in a quantum superposition of being a proton and a neutron (or something equally absurd)?

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    $\begingroup$ You might want to remove the last quip, as the isospin 0 deuteron is a singlet combination of up/down isospins: $(|pn\rangle-|np\rangle)/\sqrt 2$. $\endgroup$
    – JEB
    Mar 27, 2023 at 16:05
  • $\begingroup$ "the nature of the photon" is that of an irreversible energy exchange between the electromagnetic field and an external system. That energy exchange can never happen without a (quantized) exchange of angular momentum. So what you are actually observing is that "the absorbed photon" makes your (classical) polarizer rotate slowly. If you are working in reflection, it makes it rotate twice as fast (because now you are looking at an absorption/emission process). Since you are in charge of which way the polarizer can rotate, you can change the polarization part of the wave function. $\endgroup$ Mar 27, 2023 at 16:53

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Linear polarization is a tensor alignment, so the basis should be written with double arrows (an alignment lacks a direction):

$$ \{|\leftrightarrow\rangle, |\updownarrow\rangle\} $$

These represent field alignments that are orthogonal to the propagation direction, $\hat k$. So for each $\vec k$ there are two independent states.

Also, up to a phase convention, the state in your question is an eigenstate of circular polarization (which is a vector polarization, but since it's constrained to be transverse you can label it by helicity):

$$ |+\rangle = \frac 1 {\sqrt 2}\big(|\leftrightarrow\rangle+ |\updownarrow\rangle\ \big)$$

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