For context, this is from Page 27 of Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press:
She then sends the first qubit through a Hadamard gate, obtaining:
This state may be re-written in the following way, simply by regrouping terms:
This expression naturally breaks down into four terms. The first term has Alice’s qubits in the state |00〉, and Bob’s qubit in the state α|0〉 + β|1〉 – which is the original state |ψ〉. If Alice performs a measurement and obtains the result 00 then Bob’s system will be in the state |ψ〉. Similarly, from the previous expression we can read off Bob’s post- measurement state, given the result of Alice’s measurement:
Depending on Alice’s measurement outcome, Bob’s qubit will end up in one of these four possible states.
Regrouping the terms in expression 1.31 in Quantum Computing and Quantum Information, Nielsen and Chuang
I understand how the calculations are being done here, but I still don't understand as how did the information regarding the transferred qubit reach Bob. What physical significance does the distributive law have? Through this regrouping, how did we manage to influence Bob's qubit?
What exactly is taking place? To use a classical analogy, let us say that A has 2 containers, one with x balls and the other with either 0 or 1 ball, A knows that if A's box has 0 balls then B's box must also have 0 balls (same for 1).
Now, since we have a total of either (x+0+0) balls or (x+1+1) balls, we also have (0+0+x) balls or (1+0+[x+1]) balls. Voila, B now has a different number of balls in his box! If we open our first box and see 0 balls, then B's box has x balls! So what's happening here physically?
