von Neumann  and Gleason vs. Bell

Returning to physics topics today I want to talk about an important contention point between von Neuman and Gleason on one had, and Bell on the other. I had a series of posts about Bell in which I discussed his major achievement. However I do not subscribe to his ontic point of view and today I will attempt to explain why and perhaps persuade the reader with what what I consider to be a solid argument. 

Before Bell wrote his famous paper he had another one in which he criticized von Neumann, Jauch and Piron, and Gleason. The key of the criticism was that additivity of orthogonal projection operators not necessarily implies the additivity of expectation values:

\(\langle P_u + P_v \rangle = \langle P_{u}\rangle + \langle P_{v}\rangle \)

The actual technical requirements in von Neumann and Gleason case were slightly different, but they can be reduced to the statement above and more importantly this requirement is the nontrivial one in a particular proof of Gleason's theorem. 

Andrew Gleason

To Bell, additivity of expectation values is a non-natural requirement because he was able to construct hidden variable models violating this requirement. And this was the basis for his criticism of von Neumann and his theorem of the impossibility of hidden variables. But is this additivity requirement unnatural? What can happen when it is violated? I will show that violation on additivity of expectation values can allow instantaneous communication at a distance.

The experimental setting is simple and involves spin 1 particles. The example which I will present is given in late Asher Peres book: Quantum Theory: Concepts and Methods at page 191. (This book is one of my main sources of inspiration for how we should understand and interpret quantum mechanics. )

The mathematical identity we need is:

\(J_{z}^{2} = {(J_{x}^{2} - J_{y}^{2})}^2\)

and the experiment is as follows: a beam of spin 1 particles is sent through a beam splitter which sends to the left particles of eigenvalue zero for \(J_{z}^{2}\) and to the right particles of eigenvalue one for \(J_{z}^{2}\).

Now a lab on the right decides to measure either if \(J_z = 1\) or if \(J_{x}^{2} - J_{y}^{2} = 1\)

For the laboratory on the right let's call the projectors in the first case \(P_u\) and \(P_v\) and in the second case \(P_x\) and \(P_y\)

For the lab on the left let's call the projectors in the first case \(P_{w1}\) and in the second case\(P_{w2}\). 

Because of the mathematical identity: \(P_u + P_v = P_x +P_y\) the issues becomes: should the expectation value requirement hold as well?

\(\langle P_{u}\rangle + \langle P_{v}\rangle  = \langle P_{x}\rangle + \langle P_{y}\rangle \)

For the punch line we have the following identities:

\(\langle P_{w1}\rangle = 1 - \langle P_{u}\rangle - \langle P_{v}\rangle\) 

and

\(\langle P_{w2}\rangle = 1 - \langle P_{x}\rangle - \langle P_{y}\rangle\) 

and as such if the additivity requirement is violated we have:

\(\langle P_{w1}\rangle \neq \langle P_{w2}\rangle\)

Therefore regardless of the actual spatial separation, the lab on the left can figure out which experiment the lab on the right decided to perform!!!

With this experimental setup, if additivity of expectation values is false, you can even violate causality!!!

Back to Bell: just because von Neumann and Gleason did not provide a justification for their requirements, this does not invalidate their arguments. The justification was found at a later time. 

But what about the Bohmian interpretation of quantum mechanics? Although there are superluminal speeds in the theory, superluminal signaling is not possible in it. This is because Bohmian interpretation respects Born rule which is a consequence of Gleason't theorem and it respects the additivity of  expectation values as well. Bohmian interpretation suffers from other issues however.