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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 a 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 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 seem 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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    $\begingroup$ May be tangential to your question, but here's another question about the same article: physics.stackexchange.com/q/17170/3998 $\endgroup$
    – Siva
    Mar 20, 2013 at 5:37
  • $\begingroup$ 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? $\endgroup$
    – user154997
    Aug 30, 2017 at 10:21
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    $\begingroup$ 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). $\endgroup$ Aug 25, 2020 at 13:05
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    $\begingroup$ Isn't this a "check-my-work" question? $\endgroup$
    – Jim
    Sep 14, 2020 at 14:02

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Most of this is technically correct (not sure about the first equation) just phrased in a misleading way. Scaling is important when quantities tend to be very large, or very small. In this case the relevant quantities are O(1) so the required number of measurements is not "bad".

For example, if we choose $\theta = \pi / 8$, we need only $n = 4$. Then a confidence of $\epsilon = 0.01$ would require $N \simeq 10^4$. For an experiment with photons this is a small number.

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