# Why does the Pusey-Barrett-Rudolph (PBR) theorem focus on one-to-many relationships?

The PBR theorem says that one state of a hidden variable theory cannot correspond to multiple pure states of Quantum Mechanics.

But, wouldn't hidden variable theories be of a many-to-one nature anyway? This is because hidden variables are supposed to provide a complete information of reality, while the pure state wavefunction is supposed to be incomplete information. Then, doesn't this mean than we should lose information when switching from the hidden variable description to the wavefuncteion description?

That would mean that multiple hidden variable states should correspond to the same pure state. I think this is what the case with Bohmian mechanics is. Bohmian mechanics needs the wavefunction as well as a hidden variable to specify the full state. This means that multiple hidden variable states can share the same wavefunction.

So, why is the PBR theorem about one-to-many relationships rather than many-to-one?

I would suggest simply reading the original PBR paper "On the reality of the quantum state" - the introduction is highly readable and explains the kind of hidden variable model their theorem disproves very well. I'm just paraphrasing them in the following:

In order to account for the varying results of measurements on identically prepared quantum systems in "the quantum state" $$\psi$$, a hidden variable theory needs to associate a spread of hidden variable states to that method of preparation, i.e. a distribution $$\mu_\psi(\lambda)$$ over the actual physical states $$\lambda$$ created by that preparation.

Now, when you have two different preparations that result in quantum states $$\psi_1$$ and $$\psi_2$$, one can ask whether the supports of the corresponding hidden variable distributions $$\mu_{\psi_1}$$ and $$\mu_{\psi_2}$$ are disjoint or not. If they are not, then there is at least one "true" physical state $$\lambda$$ that is realized by both preparations, and hence the quantum states $$\psi_i$$ are not an "ontic" aspect of the physical states in the support of the hidden variable distributions.

In order to understand what this means and why this is interesting, PBR draw the analogy to classical statistical mechanics, where you might have different distributions $$\mu_E(x,p)$$ on phase space indexed by some parameter $$E$$. When they don't overlap (such as e.g. the microcanonical macrostates for different energies $$E$$), then the thing that labels these non-overlapping distributions (e.g. the different energies) can be seen as a shared physical property of all physical states in the support of the distribution, i.e. the family of distributions $$\mu_E(x,p)$$ induces a well-defined map that associated to each actual physical state $$(x,p)$$ its corresponding property $$E$$ by choosing the one $$E$$ where it is in the support of $$\mu_E(x,p)$$.

When the distributions $$\mu_E$$ overlap, this map is no longer well-defined since states can be in the support of more than one $$\mu_E$$, and this is the one-to-many mapping the PBR theorem is concerned with.

Coming back to the quantum theory, the PBR theorem shows that the distributions $$\mu_{\psi_1}$$ and $$\mu_{\psi_2}$$ cannot overlap if the hidden variable theory is supposed to reproduce the predictions of quantum mechanics, and that hence quantum states are as meaningful a property of the underlying physical states as energy is in classical statistical mechanics, in particular to each physical state there is a unique assignment of a corresponding quantum state.

• Do you know if the Arxiv version (arxiv.org/abs/1111.3328) differs substantially from the final published version? I'm trying to digest the Arxiv version (and frankly finding myself disagreeing with its assumptions for the same reason as the OP) but unsure if I'm reading a complete version of the argument. Thanks! Dec 11, 2022 at 21:22