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Hilbert space and its different sets of orthonormal bases (ONBs) provide an excellent framework for describing complementarity. In spin-1/2 space we chose the $(z^+,z^-)$ basis and make measurements. After the measurement we obtain objects whose properties wrt the different ONBs $(y^+,y^-)$ and $(x^+,x^-)$ and others are maximally undefined. There are many more examples in higher dimensional scenarios, as well, and all of this is supposed to be well understood.

Question: Why do we always end up with the same number of dimensions? Why is not that choosing basis $(z^+, z^+)$ leaves us with some maximally undefined properties for whose understanding we need, say, a 4-dimensional space with a basis, say, $(a,b,c,d)$?

Note 1: I am not asking why all ONBs in a single Hilbert space have the same number of elements. This is a mathematical fact and as such can be demonstrated by a mathematical proof from the relevant definitions and axioms. I am rather interested to learn why nature is such that a complementary aspect to the once chosen always leads to the same Hilbert space with the same number of dimensions.

Note 2: I realize that "Why is nature so-and-so?" are futile. We do not know. However such a question does makes sense in the following regard: Do we have any experiments or "smoking guns" which make it reasonable to expect certain features? For example: $SU(2)$ being a transformation group on qubit/spin-1/2 space and being the universal cover of $SO(3)$ we do have a "smoking gun" for expecting 3-dimensional space.

Note 3: There is a beautiful theorem by Pia Maria Soler that orthomodular lattices with (quite) some additional structure require real, complex or quaternionic Hilbert space to be implemented. Mathematically this is nice. Physically this makes very much sense: We have lots of experiments which point us to orthomodular lattices. In many experiments nature points out that property lattices should not be modeled as distributive lattices but rather as orthomodular. However I still lack the "smoking gun" forcing the same number of dimensions.

I have a mild suspicion that the answer might be connected with degeneracy of eigenvalues (we in fact have situations with different dimension and nature forces us to use, say, 6 dimensions in one scenario, while in another we might only need, say, 5 dimensions, and this is what produces a degenerate eigenspace in the latter situation where we have "too many" degrees of freedom). However as a mathematician I lack practical examples of degeneracies and thus cannot test my suspicion.

Or, is there another aspect I have completely overlooked or misunderstood?

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    $\begingroup$ when you talk of "ONB" are you referring to mutually unbiased bases? If so, I would say the reason is that MUBs are all simply different ways to "look" at the system, and as such they should all capture the same amount of information. If some basis had lower dimensionality than another, then it would correspond to a different number of possible measured states and therefore a smaller/larger information content $\endgroup$ – glS Nov 5 '18 at 14:01
  • $\begingroup$ MUBs are a typical example for this. And while your comment is very helpful, I do not yet completely see your point with the information content. If it is only about information content, a different probability distribution on a smaller set of choices could still lead to the same overall information content. $\endgroup$ – Nobody-Knows-I-am-a-Dog Nov 7 '18 at 12:03
  • $\begingroup$ so you are thinking of something more general than MUBs, or just MUBs? I ask because you say that measuring in one basis gives maximum uncertainty in another basis, which is the defining characteristic of MUBs. Also, the number of dims is essentially the number of (real) numbers that are necessary to fully characterise a state of the system. If an object is defined by its color, height and weight, then it doesn't matter how you measure these properties, you will always need to give me three "numbers" to provide a full description of said object. I don't see why complementarity is relevant here $\endgroup$ – glS Nov 7 '18 at 12:14
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    $\begingroup$ It's not correct to say that "you need different frames for description". You need just one frame/basis to characterise the system. If you change the basis/frame you are describing the same object from a different perspective, but the object that you are describing does not change and therefore requires the same amount of information to be described. It is exactly like trying to describe a vector: even if you change the basis, the underlying object that you are describing does not change, and thus requires the same amount of information/dimensions to be characterised $\endgroup$ – glS Nov 7 '18 at 12:42
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    $\begingroup$ I don't get the comment about combining 3 real values into one. If you do that you have to increase the number of bits necessary to characterise the single number, so nothing changes. This is the same as describing a 3-qubit system as a single-qudit system of dimension $2^3$ $\endgroup$ – glS Nov 7 '18 at 12:44

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