[Physics] The complementary variable to the qubit and spin-1/2


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

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?

Best Answer

"Composite" observables such as the angular momentum have no "unique" complementary observables. In fact, the three components of the angular momentum are not really "independent" in the sense of spanning a proper configuration space. Only two functions of the three angular momentum components - conventionally $j^2$ and $j_z$ - may be selected into a basis of mutually commuting observables.

Once you have this basis, you may talk about other observables that don't commute with them. There are many. Of course, they include the other components, e.g. $j_x, j_y$, of the angular momentum. If you want a treatment that is analogous to the treatment of the momenta and positions, of course that the angles $\theta,\phi$ are the natural dual variables to $j,m$. You may either use the basis of spherical harmonics, $Y_{lm}$, or you may choose the continuous basis of delta-functions located at particular values of $(\theta,\phi)$.

Clearly, for internal spin - especially the half-integer-valued spin - there is no orbital rotation so there is no $(\theta,\phi)$ basis of the Hilbert space.

The reason why there's no unique answer is that the spin is "composite" and the full Lagrangian can't be written as a function of the angular momentum only - and even if it could, there are many ways to do so. In particular, a radial motion away from the origin carries $\vec j=0$ but it is still nontrivial. For such motion, the parameterization via the angular momentum would go singular. You would need both the angular momentum and the ordinary one - but then the variables would be redundant.

For higher-dimensional space, it becomes even more clear that the angular momentum cannot describe a proper basis of the configuration or phase space because the angular momentum has many components - $d(d-1)/2$ of them - which becomes (much) higher than the actual number of components describing the motion of a point-like particle, $d$.

To summarize, the basic assumption that there exists "the" dual variable for an arbitrary observable you choose in any theory, is incorrect. Nevertheless, if you want the most accurate analogy of the relationship of the momentum- and position-basis, it's the basis of spherical harmonics and the delta-functions on the sphere and it only works for orbital angular momentum.

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