Gleason's Theorem

It feels good to be back to physics, and as a side note going forward I will do the weekly posts on Sunday. Today I want to talk about Gleason's theorem. But what is Gleason's theorem?

If you want to assign a non-negative real valued function \(p(v)\) to every vector v of a Hilbert space H of dimension greater than two, then subject to some natural conditions the only possible choice is \(p(v) = {|\langle v|w \rangle |}^{2}\) for all vectors v and an arbitrary but fixed vector w.

Therefore there is no alternative in quantum mechanics to compute the average value of an observable A the standard way by using:

\(\langle A \rangle = Tr (\rho A)\)

where \(\rho\) is the density matrix which depends only on the preparation process. 

Gleason's theorem is rather abstract and we need to unpack its physical intuition and the mathematical gist of the argument. Physically, Gleason's theorem comes from three axioms:

• Projectors are interpreted as quantum propositions
• Compatible experiments correspond to commuting projectors
• KEY REQUIREMENT: For any two orthogonal projectors P, Q, the sum of their expectation values is the expectation value of P+Q: \(\langle P \rangle + \langle Q\rangle = \langle P+Q\rangle\)

In an earlier post I showed how violating the last axiom (which is the nontrivial one), in the case of spin one particles, can be used to send signals faster than the speed of light and violate causality. But how does Gleason arrives at his result?

Let's return at the original problem: to obtain a real non-negative function p. Now add the key requirement and demand that for any complete orthonormal basis \(e_m\) we have:

\(\sum_m p(e_m) = 1\)

For example in two dimensions on a unit circle we must have:

\(p (\theta) + p(\theta + \pi/2) = 1\)

which constrain the Fourier expansion of \(p (\theta)\) such that only components 2, 6, 10, etc can be non zero. In three dimensions the constraints are much more severe and this involves rotations under SO(3) and spherical harmonics. I'll skip the tedious math, but it is not terribly difficult to show that the only allowed spherical harmonics must be of order 0 and 2 which yields: \(p(v) = {|\langle v|w \rangle |}^{2}\).

The real math heavy lifting is on dimensions larger than three and to prove it Gleason first generalizes  \(\sum_m p(e_m) = 1\) to \(\sum_m f(e_m) = k\) where k is any positive value. He names this "f" a "frame function". Then he proceeds to show that dimensions larger than three do not add anything new.

If you are satisfied with the Hilbert spaces of dimension 3, the proof of the theorem is not above undergrad level, and I hope it is clear what the argument is. But what about Many Worlds Interpretation? Can we use Gleason's theorem there to prove Born rule? Nope. The very notion of probabilities is undefined in MWI, and I am yet to see a non-circular derivation of Born rule in MWI. I contend it can't be done because it is a mathematical impossibility and I blogged about it in the past.