An old friend of mine, Jacques Pienaar, wrote to me last Friday asking whether this paper by Wang et al. is bullshit, or is he going crazy. Don’t worry, Jacques, you’re fine. The paper is bullshit.
It claims to experimentally demonstrate the violation of a Bell inequality using unentangled photons. Which is of course impossible. It’s a simple and well-known theorem that separable states cannot violate a Bell inequality. Let me prove it here again for reference: let $\rho_{AB} = \sum_\lambda p_\lambda \rho^A_\lambda \otimes \rho^B_\lambda$ be a separable state shared between Alice and Bob, who measure it using POVMs $\{A^a_x\}_{a,x}$ and $\{B^b_y\}_{b,y}$. Then the conditional probabilities they observe are
\begin{align*}
p(ab|xy) &= \tr[\rho_{AB} (A^a_x \otimes B^b_y)] \\
&= \sum_\lambda p_\lambda \tr(A^a_x \rho^A_\lambda) \tr(B^b_y \rho^B_\lambda) \\
&= \sum_\lambda p_\lambda p(a|x \lambda) p(b|y \lambda),
\end{align*} so we directly obtain a local hidden variable model for them, with hidden variables $\lambda$ distributed according to $p_\lambda$, with response functions $p(a|x\lambda)$ and $p(b|y \lambda)$. Therefore they cannot violate any Bell inequality.
The authors certainly know this. Well, I know Mario Krenn and Anton Zeilinger personally, and I know that they know. The others I can safely presume. This means that the paper is not merely mistaken. It is bullshit.
But what have they done?, you ask. How have they obtained the result they claim? Honestly, I don’t know, and it’s not my problem. When you contradict a well-known theorem you’re the one who has to explain why the theorem is wrong, or why it doesn’t apply to your situation. They don’t explain it. I read the paper, and all they have to say on this subject is the first sentence of the abstract “Violation of local realism via Bell inequality […] is viewed to be intimately linked with quantum entanglement”. They also talk about “the ninety-year endeavor in the violations of local realism with entangled particles.” Apparently it’s not a theorem, just an opinion? Or tradition?
So I’m perfectly justified in washing my hands. I can’t contain my curiosity, though. Is the quantum state they used actually entangled? Or is the violation of a Bell inequality just fake? The state does seem to be separable, so there’s only option left: there is no violation. Their setup is inherently probabilistic, they use four photon sources, which can in total generate either 0, 2, 4, 6, or 8 photons. They postselect on the detection of four photons. Well, well. It’s well-known that you can fake a violation of a Bell inequality via post-selection, if the detection efficiency is $\le 2/3$. What is their efficiency? They don’t reveal. But they do seem to be aware that their experiment is not halal, as they write “We expect that tailored loopholes and local hidden variable to the work reported here can be identified.”
UPDATE: Wharton and Price just uploaded a comment to the arXiv, confirming that the Bell violation is faked through postselection.
UPDATE 2: Now Cieśliński et al. uploaded another comment to the arXiv. They agree with Wharton and Price’s analysis, and additionally show that one can make their experimental setup produce a real Bell violation by adding on/off switches to optionally block the pumping field going to the second downconverters. But then they become part of the measurement device, not state preparation, and the state at this point is entangled.
I am bothered by the language of this comment. Which does not even call itself a comment. Despite refuting everything about the paper, they call it “brilliant” and “outstanding”. They also can’t even bring themselves to directly say the obvious, that a Bell violation with separable states is impossible. They write that “Moreover, we provide an analysis which shows that the statement in their title, about unentangled photons, cannot be upheld.” They also repeatedly talk about the “conjecture” by Wang et al. that they violated a Bell inequality. That’s ridiculous, they didn’t “conjecture” anything, they directly and repeatedly claim a Bell violation. I hope they grow some self-respect and that in the next version of the comment they use proper scientific language.
UPDATE 3: Sabine Hossenfelder made a YouTube video about the paper. I find it mysterious why she should care about it, since she’s a superdeterminist. In any case, the video is embarrassing. Doubly so because it will have orders of magnitude larger audience than the comments. She gives the paper a “0 out of 10 in the bullshit meter”, dismisses the correct explanation of postselection by demonstrating she has no idea what postselection is, and concludes with “of course there’s always the possibility that I just don’t understand it”. Indeed, Dr. Hossenfelder, on this point you are correct.