# Measuring a state $\frac{1}{2}|0\rangle-\frac{\sqrt 3}{2}|1\rangle$ in the $X$ and $Z$-bases?

If a qubit is in the state $$|\psi\rangle = \frac {1}{2}|0\rangle - \frac{\sqrt 3}{2} |1\rangle$$, how do I measure it in the $$Z$$-basis, i.e. $$\{|0\rangle,|1\rangle\}$$, and the $$X$$ basis, i.e. $$\{|+\rangle,|-\rangle\}$$, and find the states and their probabilities?

What I thought of doing is for $$\{|0\rangle,|1\rangle\}$$ is: I take the squared absolute value of $$\alpha$$ and $$\beta$$ without any modification or conversion, and that would be my probabilities.

But I'm confused on what to do with $$\{|+\rangle,|-\rangle\}$$?

• This sounds like homework, but you've copied the question in a way that doesn't make any sense, and you've not indicated what effort, if any, you have already taken. For example, it's clear that you intended to indicate the X basis in question (b). Also, your state is not normalized, it's not clear what the amplitudes for |0⟩ and |1⟩ are, because amplitudes of 21 and 23 don't make any sense. Please consider initially reviewing your question properly, and editing your question to include the details of what you've already done. – Commented Feb 16, 2022 at 21:30
• Good job on slowly turning your question around. I've retracted my vote to close. Commented Feb 16, 2022 at 22:22

Your approach to solving the case in the $$Z$$-basis $$\{|0\rangle, |1\rangle\}$$ is correct.

You can reduce the other case to the one you already know how to solve by expressing $$|\psi\rangle$$ in the $$X$$-basis $$\{|+\rangle, |-\rangle\}$$. To that end, you can use

$$|0\rangle=\frac{|+\rangle+|-\rangle}{\sqrt2}\\ |1\rangle=\frac{|+\rangle-|-\rangle}{\sqrt2}.$$

• what about if I need it in z-basis?
– n22
Commented Feb 16, 2022 at 23:25
• I am a beginner and I tried conversion this way : (1/2 + - 3^(1/2) / 2) / 2^(1/2) |+> + (1/2 + - 3^(1/2) / 2) / 2^(1/2) |-> and I keep getting 2+ 3^(1/2) / 4
– n22
Commented Feb 16, 2022 at 23:28

You can apply a Hadamard gate to your state, and then proceed to measure in the computational basis