In their recent paper The Quantum State Can Be Interpreted Statistically, Lewis et al. end with a very nice mathematical question, one whose answer (either way) would have interesting implications for the foundations of quantum mechanics. In the hopes of getting some non-quantum math folks interested in their question---and maybe even finding someone to say "the answer is trivial for the following reason..." :-)---I decided to state the question for the MO community, shorn of all the physics and philosophy.

Let H_{d} be the set of unit vectors in $\mathbb{C}^d$. A *ψ-epistemic theory* in d dimensions consists of the following:

- A measurable space Λ (called the "space of ontic states").
- A function mapping each unit vector ψ∈H
_{d}to a probability measure D_{ψ}over Λ. - A function f(λ,M,i)∈[0,1], which takes as input an ontic state λ∈Λ, an ordered orthonormal basis M=(v
_{1},...,v_{d}) for $\mathbb{C}^d$, and an index i∈{1,...,d}.

f must satisfy the following two conditions:

(i) $\sum_{i=1}^{d}f(\lambda,M,i)=1$ for all λ and M. (Intuitively, f must give rise to a probability distribution over the "measurement outcomes" v_{1},...,v_{d} in M.)

(ii) $\int_{\lambda \sim D_{\psi}} f(\lambda,M,i) d\lambda = |v_{i}^{*}\psi|^{2}$ for all ψ, M, and i. (Intuitively, the probability of the measurement outcome v_{i}, averaged over all λ drawn from D_{ψ}, must equal the squared projection of ψ onto v_{i}.)

Note that we can trivially satisfy conditions (i) and (ii) as follows:

- Λ=H
_{d} - D
_{ψ}assigns probability 1 to λ=ψ, and probability 0 to all other states in Λ - f(ψ,M,i) = |v
_{i}^{*}ψ|^{2}

Thus, let Supp(D)⊆Λ be the support of D, and call a ψ-epistemic theory *nontrivial* if there exist ψ≠ϕ such that $Supp(D_{\psi})\cap Supp(D_{\phi}) \ne \emptyset$.

Observe that, if ψ and ϕ are orthogonal, then Supp(D_{ψ}) and Supp(D_{ϕ}) must be disjoint. This is because, if we set v_{1}=ψ and v_{2}=ϕ, then $v_{1}^{*}\psi = v_{2}^{*}\phi = 1$ and $v_{1}^{*}\phi = v_{2}^{*}\psi = 0$, which is not possible if D_{ψ} and D_{ϕ} have any nonzero overlap. Motivated by this observation, call a theory *maximally nontrivial* if $Supp(D_{\psi})\cap Supp(D_{\phi}) \ne \emptyset$ whenever ψ and ϕ are *not* orthogonal.

I can now state Lewis et al.'s open problem:

Does there exist a maximally-nontrivial ψ-epistemic theory in dimensions d≥3?

**Update: See the comments for an extremely nice solution by George Lowther, plus my followup questions.**

I know of two results directly relevant to this problem.

First, there exists a maximally-nontrivial theory in dimension d=2, which was found by Kochen and Specker in 1967. See this paper by Rudolph for more details, including why the obvious generalizations to 3 or more dimensions seem to fail. Briefly, the Kochen-Specker theory is defined as follows:

- Λ=H
_{2}. - D
_{ψ}assigns probability measure $2 | \psi^{\*} \phi|^{2} - 1$ to ϕ if $| \psi^{\*} \phi|^{2} \geq 1/2$, and probability measure 0 to ϕ otherwise. - f(ψ,M,i) = 1 if $|v_{i}^{\*} \psi|^{2} \geq 1/2$, and f(ψ,M,i) = 0 otherwise.

(Warning: I converted from a different representation, and can't promise I didn't get a factor of 2 wrong or something like that.)

The second result is that, for all finite d, there exists a nontrivial ψ-epistemic theory (though it's far from being *maximally* nontrivial). This is the main result of Lewis et al.

My own guess is that maximally-nontrivial theories *don't* exist for d≥3, but I'd only give it 60% confidence.

To anticipate some questions:

Yes, I'd also be interested in this problem with $\mathbb{R}$ in place of $\mathbb{C}$ (though I suspect the two cases are pretty similar).

Yes, I'd be interested in negative results for restricted classes of theories. Here are a few examples of restrictions one could look at, in various combinations: Λ=H

_{d}, f∈{0,1}, f is continuous, symmetry under unitary transformations, symmetry under relabeling of the v_{i}'s.No, I don't know how to rule out that the answer could depend on the Axiom of Choice or something crazy like that (but I doubt it).

**Update (March 20, 2013):** Adam Bouland, Lynn Chua, George Lowther, and myself now have a paper on ψ-epistemic theories originating with this MO post. The paper contains the construction below, but also proves impossibility results for ψ-epistemic theories when an additional symmetry condition is imposed.

infiniteconvex combination, to handle the limit as $|ψ^{∗}ϕ|$ approaches 0, since clearly we'll always have $\epsilon \lt |ψ^{∗}ϕ|$). – Scott Aaronson May 1 '12 at 3:00