QUOTE="Eye_in_the_Sky, post: 5550830, member: 11112"]
i) If Bob's setting had been b2 instead of b1, each of Alice and Bob would have obtained a definite outcome.[/QUOTE]
That statement is true, depending on what you mean by 'definite outcome'.
Alice and Bob perform measurements - they get definite outcomes
(They get a definite result - just not a
predictable result)
That's true whatever spin direction they each choose. The outcomes are only perfectly correlated if they happen to choose the same spin observable to measure.
Let's suppose they decide to do the following. They decide to do the first 10,000 runs of the experiment both measuring spin-z, the next 10,000 with Alice measuring spin-z and Bob measuring spin-(z + a), and so on. As Bob keeps changing his setting progressively more towards the spin-x direction, they will see a correspondingly weaker degree of correlation between their results until when Bob reaches spin-x there is no correlation between their results whatsoever (in the limit of a very large number of experimental runs, of course - for 10,000 runs there is a very, very, very small probability that they will obtain perfectly correlated results even with spin-z and spin-x measured).
So even if we have a perfectly entangled state there will be no correlation between the
observables spin-z and spin-x. So, clearly, there's something a bit more going on than just straightforward correlation between observable pre-existing 'properties'.
In my view a fundamental parameter is the mutual information (we called it the 'index of correlation' in our stuff, but it's just essentially known as the entropy of entanglement these days). This is just I = S(A) + S(B) - S, where S(A), S(B) are the entropies of the reduced systems and S is the total entropy. In effect this is the difference in information between examination of the separate systems alone and examination of their joint properties.
For classical systems this quantity is less than or equal to inf [ S(A), S(B) ]. For quantum systems this quantity is less than or equal to 2 inf [ S(A), S(B) ]. So in other words the correlation (as parameterized by the mutual information) can be twice as large as the corresponding 'equivalent' classical system. This comes about because of the Araki-Lieb inequality that yields the quantum result
| S(A) - S(B) | ≤ S. It's the possibility of pure entangled states that gives this rather surprising and beautiful relation.
In a very hand-waving kind of way it's this 'extra' information inherent in entangled states that gives rise to the various things we can do with entanglement - like dense-coding, teleportation, entanglement-swapping, etc.
