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
• a dense and rigourous review of the background of the PBR theorem . – user46925 Dec 30 '15 at 20:14
• 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

Aaah, the theory that "shook the foundations"... by not shaking anything in particular, too much fuss about nothing honestly.

The theory tried to refute some interpretations of QM that claimed "secret" quantum states, that are essentially different than wave-functions we are accustomed to use in order to describe the physical state of the system. In other words they refuted the interpretations that claimed that these "secret states" do not determine the wave-functions.

Therefore the theory disproved straw-men... or should I say "the type of QM interpretations you wouldn't read about in a book but you could find them on certain websites".

There is no experimental evidence for or against it, it would be very costly to make such experiments.

protected by Community♦Jan 12 '17 at 3:13

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