Context: The recent paper The quantum state cannot be interpreted statistically by Pusey, Barrett and Rudolph shows under suitable assumptions that the quantum state cannot be interpreted as a probability distribution over hidden variables (go read it now).

In the abstract they 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 starting with: "In a real experiment, it will be possible to establish with high condence 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 suprising, I suppose) *and* the error tolerance (bad, right?).

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