This is a subtle sort of thing that often leads to papers that make correct statements in the body of the paper but have titles and abstracts alluding to sending information superluminally or back in time or future events influencing the past. That's all nonsense and there is always a "trick" that makes it not what you were hoping for. Every time it makes me curious as to whether the authors realize they are pulling one over on you when they are writing the title and abstract or whether they are pulling one over on themselves without realizing it.
The question you ask is a very good example of such a thing. Yes, Alice can do a joint measurement on (a1,a2) that will cause a correlation or anti-correlation of Bob's particles (b1,b2). However, Bob won't know whether (b1,b2) are correlated or whether they are anti-correlated until Alice tells him the outcome of her measurement.
Now here is something that you may find shocking: none of this requires quantum mechanics at all! When you think you may have spotted some quantum magic, the first thing to do is substitute the words "probability distribution" wherever you used the words "quantum state", and "corellation" where you used "entanglement". Here is how it goes in the present case. Let a1 and a2 be random and independent. Let b1=a1 and b2=a2. So initially, b1 and b2 are uncorrelated. Now Alice measures whether a1 and a2 are the same. If they are, then instantaneously (faster than the speed of light!) Bob's values (b1,b2) go from being uncorrelated to being correlated! And if Alice sees that a1 and a2 are not the same, then Bob's particles instantaneously become anti-correlated.
Does this sound like magic? Probably not. Does information travel faster than the speed of light here? It depends on how you define things. But in my opinion, no. There are various tricks that sometimes get thrown into the mix. One is post-selection. The way that post-selection works is that Alice checks whether a1=a2 and you only consider the cases where that check turns up positive. This way, after Alice's measurement, Bob's particles are guaranteed to always be correlated. Post-selection has important theoretical uses in some cases. In other cases, such as in problems resembling the one here, it seems to mainly be used for confusing the audience (or even the author) in my somewhat cynical opinion.