Experimentalists usually measure in the Z basis. However, it might be useful to be able to measure in any basis, for instance, when we want to know if a qubit is in the plus state. It turns out that we can do so on a controllable qubit by first applying an operator, and then measuring in the Z basis.
- Linear algebra
- Representing qubits
Let’s try this for ourselves. Which operator U do we need to apply to a qubit to measure it in the X basis? What about the Y basis? What does the 0 state collapse to if we measure it in the X basis using our protocol?
These lecture notes from Berkeley University give a great overview on the basics of Measurements in Quantum Mechanics.