Aug 20, 2018

You are very close to the answer why his state is useless, essentially your question “you pointing out that to transfer information you need an observer?” nearly answers it. More correctly, as I tried to explain in the post, Bob needs the information of the observation outcome of Alice to know, what state he has. Note that Bob does not perform any measurement, otherwise the state which Alice wanted to transfer to him would collapse.

What you try to get at in the rest of your comment is also close to what happens — the only thing you do not take into account is that not only 0 or 1 are possible states to be transferred (which would be completely classical), but also superpositions of the two.

Let’s try to take the two ingredients I mentioned above (Bob needs Alice’s measurement outcome and the message can be a superposition) together to explain why Bob’s state is useless without the information of Alice’s measurement.

Let’s say Alice wanted to transfer the state (a|0>+b|1>), i.e. a superposition of 0 and 1. If you do the maths (which are unfortunately slightly too complicated to explain here), then you see that after Alice performs the CNOT gate and measures, Bob’s state is not always (a|0>+b|1>), but can also be (a|0>-b|1>) or (a|0>+i*b|1>) (the i is the imaginary i) or (i*a|0>+b|1>). In fact, these four combinations are* all possible combinations, *i.e. Bob only knows that he has “something”, but the “something” can be anything as far as he is concerned. It’s as if you send someone a message of 12 letters but the receiver only knows that he received 12 letters, but not which 12 letters, so the message is useless.

The cool thing is that which combination Bob actually got depends on Alice’s measurement outcome! So if Alice sends Bob her measurement, Bob can just perform a few operations on his state and then he *always* ends up with (a|0>+b|1>), i.e. the state to be transported.