# Experimental test of the non-statisticality theorem?

Context: The paper On the reality of the quantum state (Nature Physics 8, 475–478 (2012) or arXiv:1111.3328) shows under suitable assumptions that the quantum state cannot be interpreted as a probability distribution over hidden variables.

In the abstract, the authors claim: "This result holds even in the presence of small amounts of experimental noise, and is therefore amenable to experimental test using present or near-future technology."

The claim is supported on page 3 with: "In a real experiment, it will be possible to establish with high confidence that the probability for each measurement outcome is within $\epsilon$ of the predicted quantum probability for some small $\epsilon> 0$."

Something felt like it was missing so I tried to fill in the details. Here is my attempt:

First, without even considering experimental noise, any reasonable measure of error (e.g. standard squared error) on the estimation of the probabilities is going to have worst case bounded by:

$$\epsilon\geq\frac{2^n}{N},$$

(Tight for the maximum likelihood estimator, in this case) where $n$ is the number of copies of the system required for the proof and $N$ is the number of measurements (we are trying to estimate a multinomial distribution with $2^n$ outcomes).

Now, they show that some distance measure on epistemic states (I'm not sure if it matters what it is) satisfies:

$$D \ge 1 - 2\epsilon^{1/n}.$$

The point is, we want $D=1$. So, if we can tolerate an error in this metric of $\delta=1-D$ (What is the operational interpretation of this?), then the number of measurements we must make is:

$$N \ge \left(\frac4\delta\right)^n.$$

This looks bad, but how many copies do we really need? Note that the proof requires two non-orthogonal qubit states with overlap $|\langle \phi_0 |\phi_1\rangle|^2 = \cos^2\theta$. The number of copies required is implicitly given by:

$$2\arctan(2^{1/n}-1)\leq \theta.$$

Some back-of-the-Mathematica calculations seems to show that $n$ scales at least quadratically with the overlap of the state.

Is this right? Does it require (sub?)exponentially many measurements in the system size (not surprising, I suppose) and the error tolerance (bad, right?).

• May be tangential to your question, but here's another question about the same article: physics.stackexchange.com/q/17170/3998 – Siva Mar 20 '13 at 5:37
• You have both $\delta \le 2\epsilon^\frac{1}{n}$ and $4N^{-\frac{1}{n}}\le 2\epsilon^\frac{1}{n}$. Your inequality for $N$ is equivalent to $\delta$ being greater that the latter lower bound: how do you justify this? – user154997 Aug 30 '17 at 10:21
• A review of the PBR theorem and its background is in 'Is the Quantum State Real? An Extended Review of ψ-ontology Theorems', MS Leifer, Quanta 3, 67 (2014). – Emilio Pisanty Aug 25 '20 at 13:05
• Isn't this a "check-my-work" question? – Jim Sep 14 '20 at 14:02