michael_nielsen

That seems discouraging. But maybe we can tip the problem upside down? Might it be possible for Alice to do a measurement that DOESN'T acquire any information about the amplitudes a and b, but somehow can still help Bob acquire the original state?

That sounds pretty unlikely. But if you know some classical cryptography, you'll know that something like this is done routinely in the cryptosystem known as a one-time pad. Here's how it works.

Alice has a data bit, d, which she'd like to send to Bob. Alice and Bob share a pair of random bits (r, r). Alice XORs d onto her half of the random pair, and sends the result d + r to Bob. Bob can then recover d by adding (d+r) to his random bit: d+(d+r) = d (modulo 2).

The interesting thing for us is that the message d+r from Alice to Bob is completely uncorrelated with the data d. In other words, that message tells us nothing whatsoever about the identity of Alice's data. (This is also why it's cryptographically useful, of course!)

This seems promising: Alice and Bob now share an entangled quantum state with the amplitudes a and b. Can they do something so Bob ends up with the original state a|0>+b|1>?