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 ϵ of the predicted quantum probability for some small ϵ>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:
ϵ≥2nN,
(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 2n outcomes).
Now, they show that some distance measure on epistemic states (I'm not sure if it matters what it is) satisfies:
D≥1−2ϵ1/n.
The point is, we want D=1. So, if we can tolerate an error in this metric of δ=1−D (What is the operational interpretation of this?), then the number of measurements we must make is:
N≥(4δ)n.
This looks bad, but how many copies do we really need? Note that the proof requires two non-orthogonal qubit states with overlap |⟨ϕ0|ϕ1⟩|2=cos2θ. The number of copies required is implicitly given by:
2arctan(21/n−1)≤θ.
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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