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For example, do the terms 'horizontal', 'vertical', 'diagonal' and 'anti-diagonal' polarization have any relevance to the physical, quantum state of a photon, or are they simply descriptive of how one orients a polarizer when measuring states?

It seems strange that basis states of a particle would be incompatible without being orthogonal in physical space, or that there wouldn't be some relationship between the behaviour of its basis states and spatial orthogonality.

(Why seems strange? Because I'm starting to get the impression that the orthogonality of spatial dimensions and the linear, distributive way they interact is one of the most impactful fundamental phenomena, most frequent root cause of 'higher-level' phenomena in reality, and that it would be unusual if the behaviour of some physical thing could not be traced back to it. If these terms are used just because they are descriptive of the actions taken to make measurements, I would suspect there was some intermediate process between the polarizer measurement-making and the behaviour of the quantum states such that the latter could be related to spatial orthogonality.

Why seems strange (less philosophically)? Bases of spin, measured via Stern-Gerlach, behave in a compatible way with spatial dimensions. Why is this different?)

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  • $\begingroup$ this probably does not directly answer your question, but may be relevant : physics.stackexchange.com/questions/538216/… $\endgroup$
    – insomniac
    Commented Mar 31, 2020 at 22:49
  • $\begingroup$ If I understand your question "... basis states of a particle would be incompatible without being orthogonal in physical space, or that there wouldn't be some relationship between the behaviour of its basis states and spatial orthogonality..." : yes, indeed, the different linearly polarized states are related via rotations. $\endgroup$
    – insomniac
    Commented Mar 31, 2020 at 22:54
  • $\begingroup$ And the representation of the rotation operation itself on the space of photon states itself is based on the requirement that photon states polarized along $\perp$ directions (spatially) are orthogonal states; that is, $\left<\uparrow|\to\right> = 0$ $\endgroup$
    – insomniac
    Commented Mar 31, 2020 at 22:56
  • $\begingroup$ I would suspect an answer to this question could only be attempted by someone who truly considers photons as particles that propagate from here to there instead of some wave function collapse. In that case polarization states along with trajectories and exact distance/locations could be understood. $\endgroup$
    – Bill Alsept
    Commented Apr 22, 2020 at 19:02
  • $\begingroup$ See en.wikipedia.org/wiki/Bloch_sphere & en.wikipedia.org/wiki/Unpolarized_light#Poincar%C3%A9_sphere $\endgroup$
    – PM 2Ring
    Commented Oct 18 at 12:18

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It seems strange that basis states of a particle would be incompatible without being orthogonal in physical space, or that there wouldn't be some relationship between the behaviour of its basis states and spatial orthogonality.

It seems strange, but it's true. For the lucky case of the photon, the polarization does act like a 2D vector in the plane perpendicular to the photon's momentum, so orthogonal polarizations correspond to orthogonal vectors.

But that's not how it works in general. For a spin $1/2$ particle, orthogonal vectors correspond to antiparallel spins in real space. And the $1s$ and $2p$ states of hydrogen are orthogonal states, even though there's no reasonable way to define an "angle" between them in real space at all. Quantum mechanics simply obeys a new set of rules, it can't be reduced to classical mechanics in the way you're asking for.

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For example, do the terms 'horizontal', 'vertical', 'diagonal' and 'anti-diagonal' polarization have any relevance to the physical, quantum state of a photon, or are they simply descriptive of how one orients a polarizer when measuring states?

How one arranges things to measure something has to do with how that something interacts with the rest of the universe, and how something interacts with the rest of the universe is the only empirical sense that something can have a "state", so I'm not clear on what else you want. However, there is the complication that rotating the polarization results in a photon with the same polarization, except with its phase rotated by 180 degrees. Because of this, sometimes spaces other than normal Euclidean space, such as projective space, is used to represent the photon's polarization.

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