Basically, yes, but the dimensionality is really irrelevant. All important properties are following from the unitarity condition which specifies that we aren't losing any members of the quantum mechanical ensemble during its evolution.

This is a very subtle point, but in QM there isn't a difference. I think what's hard to accept about this is that when we say "this light is a superposition of horizontal and vertical" if makes it seem this is the correct or only way to see the light. But yeah, as you said, you could have picked any other basis instead. All these different ways of decomposing the light are equally "physical"- the most useful way to do it though is to use the basis that you're then going to be measuring in later. I'm not sure if that helped at all

@LookingGlassUniverse That does help a bit, thanks. The implication, though, seems to be that there is only one "real" state of the object and we just choose the "lens" through which we observe it to fit the purpose we've set. What does that mean for Copenhagen and the supposed reality of, for example, no definite position for a photon? Does it actually have a definite position and all we do is select the metaphorical angle at which we observe that position, such that it _appears_ that that is the "highest probability" position? (Or however it would be best to say that - I only have a middling layman's understanding of quantum physics.) Also, it's philosophically interesting because it gets into whether mathematics is discovered or invented. Are quantum objects "made of math" such that the infinite mathematical expressions of the object are literally what a superposition of infinite states is? Or is math merely a description of the physical world such that there are any number of ways to describe physical reality, only a few of which we have devised (some better - i.e. more accurate - than others) but none of which are physically real. (I'm very firmly on the side of the latter, BTW.)

​@@MichaelPizme: whoosh :) Although I tend to agree we are just forcing two states onto something that isn't in any state at all? I could have got this all wrong, as I said whoosh :)

@@1newme425 Just a few months ago, I saw a video that gave me a new perspective on quantum objects. (I don't remember the title but I'm almost certain it was by Float Head Physics.) The reason we have trouble understanding quantum objects like photons, electrons, and others is that they simply _cannot_ be described, _at all,_ using the concepts we have for macroscopic physics. "Particle," "wave," even "point," simply _do not apply_ and cannot even be "stretched" to make sense of quantum objects. They are something else entirely. We are comfortable with terms like "particle" and "wave" because they have become second-nature to us. However, until we devise entirely new concepts for quantum objects and those concepts become as familiar to us as "particle" and "wave" are now, we will fail to have anything resembling an intuitive grasp of what those objects are. They are _entirely_ different than what we're used to, so the words we use to describe what we're used to _cannot_ apply to, _cannot_ describe, them. Even with that being the case, however, it's easier to grasp what we're dealing with in quantum objects if we keep that fact in mind. They're not particles, not waves, but something else entirely. Saying that an electron doesn't have a definite position fits somewhat with our normal understanding of physics but doesn't accurately capture the reality because "position" just doesn't apply to something that is not a particle. The same applies for the term "wave." We need new concepts that are not like "particle" and "wave," then we need them to pass into common usage the same way that "particle" and "wave" have. Only then will we reach an "everyday" understanding of the quantum world.

Hey, great questions! Thanks for asking them! -This Is a confusing point; "up" and "down" aren't at right angles to each other, but when you translate them to the mathematical representation as vectors we treat them as orthogonal. The reason for this is that "orthogonality" in QM has a very specific meaning. If two states are orthogonal, then they are mutually exclusive. I.e. in an experiment, these vectors represent opposite outcomes. Since a stern-gerlach experiment has two outcomes: "up" and "down", the vectors for these must be orthogonal in QM. The vectors don't really represent the physical system (where up and down aren't at all at 90 degrees), but instead represent the information in a useful mathematical way. -Yes, you're right! You can make either the "a" or "b" coefficients complex! -I did a video about an actually "useful" thing you can do with a quantum computer once, if you're interested: kzbin.info/www/bejne/qnnJeKiZfdl_p9Esi=sQIBKjDQV94aYL0X Thanks again :)

@LookingGlassUniverse quick in the answers too! I should be the one thanking you, I will make sure to watch the video