# Measurement query

Just working through some more quantum information stuff, and have come accross a stumbling block - I'm clearly missing something obvious.

Consider a source emits states $|\Phi\rangle = \cos\theta |0\rangle + e^{i\phi}\sin\theta |1\rangle$ with fixed $\theta$ and random phases $\phi$, with equal probability for each phase.

How can I show that a measurement of the operator Z ($Z|0\rangle = |0\rangle , Z|1\rangle = -|1\rangle$) doesn't yield any information about the state emitted by the source?

James Jackson said:
Just working through some more quantum information stuff, and have come accross a stumbling block - I'm clearly missing something obvious.

Consider a source emits states $|\Phi\rangle = \cos\theta |0\rangle + e^{i\phi}\sin\theta |1\rangle$ with fixed $\theta$ and random phases $\phi$, with equal probability for each phase.

How can I show that a measurement of the operator Z ($Z|0\rangle = |0\rangle , Z|1\rangle = -|1\rangle$) doesn't yield any information about the state emitted by the source?

Just by computing the probability values: P(Z=0,|psi>)=<psi||0><O||psi> and P(Z=-1,|psi>)=<psi||1><1||psi> (<0|1>=0)

Seratend.

Thanks, I was pretty sure it was something simple I was overlooking - wood for the trees and all that!

This leads on to another measurement question: Suppose a source emits two states:

$$|\Phi_1\rangle = \frac{1}{\sqrt{2}}(|0\rangle + e^{i\phi}|1\rangle)$$
$$|\Phi_2\rangle = \frac{1}{\sqrt{2}}(|0\rangle - e^{i\phi}|1\rangle)$$

Where $\phi$ is an arbitary fixed phase. What measurement can be used to distinguish between the two states? They form an orthonormal set, so clearly can be distinguished, I just can't see 'how' to measure them.

James Jackson said:
Thanks, I was pretty sure it was something simple I was overlooking - wood for the trees and all that!

This leads on to another measurement question: Suppose a source emits two states:

$$|\Phi_1\rangle = \frac{1}{\sqrt{2}}(|0\rangle + e^{i\phi}|1\rangle)$$
$$|\Phi_2\rangle = \frac{1}{\sqrt{2}}(|0\rangle - e^{i\phi}|1\rangle)$$

Where $\phi$ is an arbitary fixed phase. What measurement can be used to distinguish between the two states? They form an orthonormal set, so clearly can be distinguished, I just can't see 'how' to measure them.

Just define another observable Z' with the good set of eigenvectors. I think now you are able to guess what vectors you have to choose.

Seratend.

Ah of course. I was trying to express the states as linear combinations of the eigenvectors of 'standard' operators rather than define my own.

Thanks for the pointers.

Just to confirm, would it be correct to define the measurement

$$M=|\Phi_1\rangle\langle\Phi_1 |-|\Phi_2\rangle\langle\Phi_2 |$$ to measure the states above?

(Edited to change + to -)

Last edited:
James Jackson said:
Just to confirm, would it be correct to define the measurement

$$M=|\Phi_1\rangle\langle\Phi_1 |-|\Phi_2\rangle\langle\Phi_2 |$$ to measure the states above?

(Edited to change + to -)

Yes for the eigenvalues +1 and -1 (but you are free to select others).

Seratend.