The Einstein-Podolsky-Rosen paradox

Albert Einstein was trying understand the new quantum theory. It certainly had successes in calculating the outcomes of experiments like, say, time-varying expectation values of things like $\langle S_x\rangle(t)$.

But he had a strong feeling that there was a problem with the idea of incompatible observables. Surely there must be some experiment that could measure any / all observables simultaneously.

The original Einstein, Podolsky, Rosen paper (1935) proposed a Gedankenexperiment, a scheme to measure two incompatible observables, momentum and position of a particle, simultaneously!

David Bohm (ca. 1950) came up with an equivalent scheme involving spin, which should allow the simultaneous knowledge of any two components of a particle's spin.

EPR in terms of spin

The neutral pi meson $\pi^0$ is an electrically neutral particle with total spin = 0. Occasionally such a particle spontaneously decays into an electron and a positron: $$\pi^0 \to e^- + e^+$$

  • Charge is conserved: The charges of the electron and positron sum to zero.
  • If the pion was initially at rest, linear momentum is conserved, so the total linear momentum of the the electron and positron (equal masses) is the same as the pion (at rest). So the two particles go flying off in exactly opposite directions, which is observed.

What about conservation of total angular momentum?

Assume the pion was at rest. It had total spin 0, that is $\hat S^2\ket{\psi_{\pi^0}}=0$. Classically, the particle is not spinning!.

What would you expect for $\hat S_z\ket{\psi_{\pi^0}}$?

After decay, there is an electron and a positron, but these are each spin 1/2 particles. So, what must happen in this kind of experiment?

In this kind of experiment?

In this kind of experiment?

Even if they are 1 m apart? 1 km apart?

Now, the EPR challenge to QM goes as follows:

  1. So, quantum mechanics, you claim that $[\hat S_x,\hat S_z]=-i\hbar\hat S_y\neq 0$ means one can't measure $S_x$ and $S_z$ simultaneously.
  2. OK, maybe it doesn't work to measure $S_z$, then $S_x$ on the same particle.
  3. So now, let's measure $S_x$ of the positron and $S_z$ of the electron.
  4. Since we know the measurements of $S_z$ always comes out opposite of each other, we now know that $S_z^p$ would have come out opposite of the measured $S_z^e$. That is $S_z^p=-S_z^e$. But we didn't actually have to make a measurement on the positron to know this. That is, we didn't have to "disturb" the positron to gain this knowledge.
  5. So now we know both $S_x$ and $S_z$ of the positron, and this violates the QM assumption about incompatible observables!
  6. Take that, QM!

Well, to be fair, this criticism does not undermine QM per se, but only points out that it must be possible to measure things that QM is not able to calculate. So in this view QM is not so much inadequate, as incomplete:

EPR has shown how to figure out both $S_x$ and $S_z$ of a particle. But QM doesn't have a way of calculating this.