# How to measure in any basis

**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? **

So we try to find an operator U such that U^{† }ZU equals X. It turns out that if we take U=H (the Hadamard gate), we obtain:

One way to get to this answer is to solve the following linear algebra problem:

Then solving for a, b, c and d will lead you to the Hadamard gate.

If we want to measure in the Y basis, we can solve a similar equation and end up with the following matrix:

One way to see this is that if we apply this gate to the Z basis states, we end up with the following:

The resulting states we can recognize as the eigenstates of the Y basis.

Second part of the question: **What does the 0 state collapse to if we measure it in the X basis using our protocol? **

If we measure the 0 state in the X basis it will collapse to one of the eigenstates of the X basis. So it will either collapse in the plus or minus state.