The qubit is a big topic of quantum information theory. A qubit is a single quantum bit. Physical examples of qubits include the spin-1/2 of an electron, for example, see page 39 of Preskill:
In quantum mechanics, two variables are called complementary if knowledge of one implies no knowledge whatsoever of the other. The usual example is position and momentum. If one knows the position exactly, then the momentum cannot be known at all. And to the extent that a situation can exist where we know something about both, there is a restriction, Heisenberg's uncertainty principle, that relates the accuracy of our knowledge:
$\sigma_x\sigma_p \ge \hbar/2$
where $\sigma_x$ and $\sigma_p$ are the RMS errors in the position and momentum, and $\hbar$ is Planck's constant $h$ divided by $2\pi$. The same relationship obtains for other pairs of complementary variables.
The units of $\hbar$ are that of angular momentum. Since spin-1/2 has units of angular momentum, it's natural that its complementary variable has no units. This is typically taken to be angle. That is, the usual assumption of quantum mechanics is that the complementary variable to spin is angle. For example, see Physics Letters A Volume 217, Issues 4-5, 15 July 1996, Pages 215-218, "Complementarity and phase distributions for angular momentum systems" by G. S. Agarwal and R. P. Singh, http://arxiv.org/abs/quant-ph/9606015
But at the same time, in quantum information theory, the concept of "mutually unbiased bases" has to do with complementary variables in a finite Hilbert space. The usual example of this is that spin-1/2 in the x or y direction is complementary to spin in the z direction. In other words, in quantum information theory, the usual complementary variable to spin is not taken to be angle, but instead is taken to be spin itself. For example, see J. Phys. A: Math. Theor. 43 265303, "Mutually Unbiased Bases and Complementary Spin 1 Observables" by Paweł Kurzyński, Wawrzyniec Kaszub and Mikołaj Czechlewski, http://arxiv.org/abs/0905.1723
But according to the Heisenberg uncertainty principle, spin can only be its own complementary variable if we have $\hbar=1$. Of course it's possible to choose coordinates with $\hbar=1$ -- this is common in elementary particles, but what I'm asking about is this:
Is there a compatible way to interpret the two different choices for the complementary variable to spin angular momentum? For example, can we also interpret spin as an angle?