My citation (see below) is one of many that demonstrate quantum nonlocality, that the Bell test synchronization between Alice's measurements and distant Bob's measurements occur FTL. We all "know" that such nonlocality "must" be instantaneous - there is no theoretical upper limit after all - but Bell tests don't normally attempt to quantify that speed or set a lower limit. The cited experiment does just that, a lower limit of 10,000 c given certain assumptions (and you don't have to agree with those).
Obviously: Alice's selection of a measurement basis causes (by assumption only) distant Bob's later measurement to be synchronized with Alice's results. This occurs FTL, possibly instantaneously, and further the causal direction is ambiguous. Put a different way: the relative timing of Alice's and Bob's measurements in a Bell test are irrelevant (this is consistent with any interpretation). Whatever influence occurs to make this happen, is labeled "quantum nonlocality" in the more modern usage of the term, and this is generally accepted as well.
What is not generally accepted any longer is the term "spooky action at a distance", but it still appears in the literature and bothers a lot of folks. To me, this phrase means the same as "quantum nonlocality" because it describes a key element of entanglement. In the cited article, they gave a DIFFERENT definition of "spooky action at a distance" than mine.
Hopefully nothing above is disagreeable to anyone, or any interpretational view.
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https://arxiv.org/abs/0808.3316
The experiment itself was not attempting to set a lower limit for the speed of quantum nonlocality per se, rather they attempt to set a lower limit for the speed of "quantum nonlocality assuming there is a preferred reference frame in the universe". They define this as "spooky action at a distance" but I don't believe Einstein used this term with the idea there might exist a preferred reference frame in the universe.
So that is really the source of the issue many of you are having with my citation, and I wasn't clear enough about my use of the citation to prevent your confusion and comments. I was primarily using the reference because it is the only one I have seen that specifically attempts to show the minimum speed of quantum nonlocality using a Bell test. Most Bell tests demonstrating violation of strict Einsteinian locality simply use c as a lower limit, and stop there (figuring any entanglement-related influence shown as exceeding c accomplishes the goal).
The Salart et al experiment goes much farther, although its objective was different: they wanted to show that Bell tests can rule out the existence of a "hypothetical universally privileged reference frame" given certain constraints ("that the Earth's speed in this frame is less than 10^-3 that of the speed of light"). Some Bohmian type interpretations, for example, posit such. Their experimental conclusion was that there is no such preferred reference frame, and since they equated this to "spooky action at a distance": they reject that (as several of you point out). They didn't reject that there are quantum nonlocal influences, just that they couldn't exist with a preferred reference frame.
Although they don't state this, I believe their experiment also points to a lower bound for the "speed of quantum information" (I would use the word "nonlocality" instead of "information"). From the paper: "
The violation of the Bell inequality at all times of the day allows one to calculate the lower bound for the speed of quantum information for any reference frame."
So while apologizing for the confusion I created by using this citation, I would stand by my main point: Bell tests demonstrate that quantum nonlocality is real, specifically that synchronization between Alice's measurements and distant Bob's measurements occur FTL (>c). Call it whatever you want: an influence, action at a distance, quantum information transfer, speed of entanglement... it's real and generally accepted by the community.
To paraphrase
@PeroK: sometimes words fail me.
[An additional note: I don't think their formal conclusion would be considered as strong today as when it was originally published in 2008. There is evidence from some CMBR studies that Earth might have a velocity relative to some cosmic reference frames that are in the neighborhood of 10^-3 c (or greater), which would then violate one of the assumptions for their specific conclusion.]