The difference between a mixed and a pure state is simply a difference in mathematical perspective, not a difference in the underlying physical reality.
> There is no experiment Bob can perform on his photon that will reveal to him whether or not Alice has measured her photon or not (note that this remains true EVEN AFTER he has received Alice's bits).
I'm not sure what you mean here, but a reasonable interpretation of your words would be false. You consider a situation where Bob has received bits from Alice even when she might not have actually made a measurement. If so then I guess you're saying that Alice just makes these up? If so, then Bob can certainly check to see if Alice has actually made the measurement she claims to have by making a local measurement on his photon, and this can result in an outcome that lets Bob definitively determine that Alice is lying to him. [Of course, sometimes Bob's result agrees with Alice's claimed bits; in this case, Alice has simply guessed correctly for what she will end up receiving in her (now fully determined) experiment.]
After a few curses of quantum mechanics (with the Sakurai book) and working in the field I still not sure if the press release is misleading or just wrong.
One of the important details is that Bob can't do any experiment to distinguish the photon before Alice measurement and the state of the photon after Alice measurement. If it's not possible for him to distinguish the states, has it really changed?
On the other hand, if Bobs makes a measurement after Alice measurement, the result should "agree" with the result Alice measurements. So, has something changed?
I think that the press release it's completely misleading and perhaps wrong.
There is absolutely no doubt that if the reader applies an intuitive classical interpretation of the term "state" to what physicists mean by a quantum state, then the reader will end up with some wrong impressions (and some right ones). The fact is that there are certain things that are going on that just don't map well on to normal, classical language. And this is not the readers fault; he's a normal, lay human being, and really smart people have been fighting over how to interpret the (unambiguous) experimental data for almost a century now without reaching a consensus.
Now, it would probably be better if physicists used the term "qustate" to refer to the state, and that they used this press releases, and that they said stuff like "when Alice makes her measurement, the qustate of Bob changes instantaneously". That way no one would go applying their classical intuition about what a "state" ought to be to the density matrix. Instead, they would ask "what's a 'qustate'?", and someone would say "well, it's sort of like a classical state...but it's not the same...and it's sorta like an epistemic probability distribution...but also not really...and the difference are subtle and very profound and you have to be careful and it takes a good amount of studying before you can understand." And that would be great and maybe a few more people would go actually read about it. But as it stands physicists are more than happy to allow laymen to misinterpret their terminology when it leads to their work sounding more exciting and getting more funding. There is definitely something of a Motte-and-Bailey going on. ( http://slatestarcodex.com/2014/11/03/all-in-all-another-bric... )
> On the other hand, if Bobs makes a measurement after Alice measurement, the result should "agree" with the result Alice measurements. So, has something changed?
Actually, the mere agreement of Alice and Bob's measurement doesn't really give evidence that anything on Bob's side has changed. If I jumble a blue marble and a red marble in an urn and then put them in separate brown bags and give you one at random, then when you look in your bag that will instantly determine what I'll get when I look in mine, but it didn't change anything about my bag in the normal, physical sense.
The way in which quantum mechanics differs from this situation is subtle but significant.
Yes, I think you hit the nail on the head here.
I have met many people for whom this is their day job who nonetheless don't understand this.
> Bob can certainly check to see if Alice has actually made the measurement she claims to have by making a local measurement on his photon
No, he can't. The best he can do is determine (probabilistically) that the bits Alice gave him are not the result of such a measurement. But he cannot know whether or not Alice's measurement was actually performed.
In other words, Bob cannot distinguish:
1. Alice does not perform a measurement, but rather simply pulls two bits out of a hat.
2. Alice does perform the measurement, but sends Bob two bits pulled out of a hat rather than the results of her measurement.
He cannot make this determination EVEN PROBABILISTICALLY, i.e. after aggregating many trials all prepared the same way. Because of this, it is wrong to say that Alice's measurement changed the state of Bob's photon.
Of course, Bob can distinguish between Alice sending him bits pulled out of a hat and bits resulting from her measurement, in which case he can know that Alice must have performed the measurement. But that's a red herring.
The misunderstanding you're now addressing arose because it doesn't make sense for you to say "this remains true EVEN AFTER he has received Alice's bits" if you're describing a situation where Alice is always making up the data. (Why would Bob ever look at it if it's always made up? It has nothing to do with the rest of the problem.)
With your clarification, I know understand what you mean by determining "whether or not Alice has measured her photon or not... EVEN AFTER he has received Alice's bits". But this doesn't change my previous comments on the common definition of "state".
That's certainly possible, but I wouldn't take it as a foregone conclusion. Nonetheless, I apologize if I came across as strident.
> Why would Bob ever look at it if it's always made up?
Why would he not? He doesn't know whether or not the bits are made up. Let's call them "corrupted" because it doesn't have to be intentional duplicity on Alice's part. It could be simple experimental error, or a noisy communications channel, or a MITM attack. None of these are distinguishable by Bob from Alice having not made a measurement. In any event, Bob might look because he wants to determine if the bits are corrupted. Or he might look because he thinks he's participating in the quantum teleportation protocol, notwithstanding that he's actually not because the bits are corrupted. Note that for any given bit pair he has to choose one or the other: teleport, or check for corruption. He can't do both.
It is also worth noting that even the possibility of corruption is enough to debunk the idea that Alice's measurement results in a physical change on Bob's side. Bob can tell (probabilistically) that there is corruption happening, and he can put a probabilistic upper bound on the amount of corruption that is happening, but under no circumstances can he tell which measurements are corrupted and which are not. One in four corrupted measurements will appear to be genuine by pure chance, and there's no way for Bob to know, even probabilistically, which is which.
As usual in these situations, it depends on being precise about what one is referring to with words like "state". But with the common definition -- a density matrix updated to incorporate all macroscopic data -- the state of A certainly does change following the measurement of B. It goes from fully mixed to pure.
Of course, Bob's personal epistemic state (which doesn't include data that he hasn't yet received from Alive) doesn't change.
You can try to say that Alice and Bob's photons have merely become entangled with the measuring apparatus, or some similar many-worlds-type statement, but then you'd be using "state" in a very different way than the vast majority of experimental physicists. You might think that definition better fitting with reality, but it certainly wouldn't make the NASA press release wrong.