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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?).

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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
    
a dense and rigourous review of the background of the PBR theorem . – igael Dec 30 '15 at 20:14

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