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

Question

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:
http://www.theory.caltech.edu/people/preskill/ph229/notes/chap5.pdf

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?

Answer 1

"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.

Answer 2

Intuitively, spin in the Z direction should have an angle around the Z axis as the complementary variable. Attempts to construct this "angle" operator (or "phase" operator conjugate to the number operator for harmonic oscillators) end up failing some of the desiderata, because the boundedness, continuity, and self-adjointness don't mix.

For more details, see papers by D. Pegg, S. Barnett, or J. Vaccaro for illustrations of how things fail, and various pseudo-phase operators that drop some of the desired properties.

For a classical illustration of some similar difficulties consider the difficulty in averaging bearings or other angular measurements. The standard "good" way of handling this is to treat the angles as points on the unit circle, and average them in the plane, and then project back onto the unit circle, which preserves exchange symmetry, angular translation covariance, and picks out the angle (should it exist) that minimizes the sum of the squares of angular distances to the others.

Lubos is correct in so much as the the wavefunction has no natural basis corresponding to theta and phi for half-integral values. However we can instead look at the transformation properties of the density matrix, which will also transform under the rotation group, and as integral representations only, so transform the same way as the coefficients of the spherical harmonics $Y_l^m$ for $l = 0 \dots 2j$, and $m = -l \dots l$. Evaluating these at a given $\theta$ and $\phi$ gives us densities for these continuous variables.

For spin-1/2, this is the well known Bloch sphere. Though we can't get an operator, in a very real way a combination of the X and Y operators does give a natural directional measurement (because we have well defined densities) that captures this information. Consider simultaneous weak measurement of both X and Y at the same time. The system will settle into a state on the equator, and is indeed a spin measurement. (We can alternately talk about a continuous POVM with elements along the equator). Both X and Y are involved even though they don't commute with each other. (Note also that the X basis is the Discrete Fourier transform of the Z basis, as implemented by a Hadamard operator. Compare this to the standard case of position and momentum. Y (and indeed any other equatorial basis) is just the X basis with differing phases applied in different elements, equivalent to a different Galilean boosted momentum basis to the Z position basis.) Now we have something equivalent to angle, but directly represented as a spin direction. Yes, the units don't work out, but any constant times a given operator is just as good representative of the complementary nature.

For spin-1 photons, the same thing can be done, with the Poincare sphere instead (think of circularly polarized light as $\pm$ Z, H and V and $\pm$ X, and D and A as $\pm$ Y).

I do not know the extent to which this works for higher dimensional systems such as massive spin-1 particles, or higher spins.

Answer 3

The kind of complementarity implied in mutually unbiased bases is slightly different from that of position and momentum.

We say that eigenstates of the Pauli operator $\sigma_z$ are complementary (or more properly mutually unbiased with) the eigenstates of - say - $\sigma_y$ because a quantum state prepared as an eigenstate if $\sigma_y$ has equal probabilities of being found in an eigenstate of $\sigma_z$. Mathematically $$ \vert _y\langle \pm \vert \pm \rangle_z\vert^2=\frac{1}{2}\, . $$ Here, $\vert \pm\rangle_k$ is any eigenstate of $\sigma_k$. In this sense, mutual unbiasedness generalizes the statement that, if a state is in an eigenstate of $\hat p$, then all positions $x$ are equiprobable: $\vert\langle x\vert p\rangle \vert^2=$ constant.

This definition of complementarity does not involve uncertainties (or product of uncertainties), but hinges the idea that, if all outcomes of the "complementary" observable are equiprobable, then you know nothing about this "complementary" observable.

Of course this has been generalized well beyond the $2\times 2$ Pauli matrices.