Imagine for a moment that we could distinguish between arbitrary quantum states. We’ll show that this implies the ability to communicate faster than light, using entan- glement. Suppose Alice and Bob share an entangled pair of qubits in the state (|00⟩ + √ |11⟩)/ 2. Then, if Alice measures in the computational basis, the post-measurement states will be |00⟩ with probability 1/2, and |11⟩ with probability 1/2. Thus Bob’s sys- tem is either in the state |0⟩, with probability 1/2, or in the state |1⟩, with probability 1/2. Suppose, however, that Alice had instead measured in the |+⟩, |−⟩ basis. Recall that √√ |0⟩ = (|+⟩ + |−⟩)/ 2 and |1⟩ = (|+⟩ − |−⟩)/ 2. A little algebra shows that the initial √ state of Alice and Bob’s system may be rewritten as (| + +⟩ + | − −⟩)/ 2. Therefore, if Alice measures in the |+⟩, |−⟩ basis, the state of Bob’s system after the measurement will be |+⟩ or |−⟩ with probability 1/2 each. So far, this is all basic quantum mechanics. But if Bob had access to a device that could distinguish the four states |0⟩, |1⟩, |+⟩, |−⟩ from one another, then he could tell whether Alice had measured in the computational basis, or in the |+⟩, |−⟩ basis.
I thought if Alice measures in +/- state, then Bob would immediately see his qubit to be in +/- state and know that she measured in +/- basis. What exactly is measuring in different bases mean then?
