Measurement query

  • #1
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 [itex]|\Phi\rangle = \cos\theta |0\rangle + e^{i\phi}\sin\theta |1\rangle[/itex] with fixed [itex]\theta[/itex] and random phases [itex]\phi[/itex], with equal probability for each phase.

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

Answers and Replies

  • #2
318
1
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 [itex]|\Phi\rangle = \cos\theta |0\rangle + e^{i\phi}\sin\theta |1\rangle[/itex] with fixed [itex]\theta[/itex] and random phases [itex]\phi[/itex], with equal probability for each phase.

How can I show that a measurement of the operator Z ([itex]Z|0\rangle = |0\rangle , Z|1\rangle = -|1\rangle[/itex]) 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.
 
  • #3
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:

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

Where [itex]\phi[/itex] 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.
 
  • #4
318
1
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:

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

Where [itex]\phi[/itex] 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.
 
  • #5
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.
 
  • #6
Just to confirm, would it be correct to define the measurement

[tex]M=|\Phi_1\rangle\langle\Phi_1 |-|\Phi_2\rangle\langle\Phi_2 |[/tex] to measure the states above?

(Edited to change + to -)
 
Last edited:
  • #7
318
1
James Jackson said:
Just to confirm, would it be correct to define the measurement

[tex]M=|\Phi_1\rangle\langle\Phi_1 |-|\Phi_2\rangle\langle\Phi_2 |[/tex] to measure the states above?

(Edited to change + to -)

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

Seratend.
 

Related Threads on Measurement query

  • Last Post
Replies
0
Views
1K
Replies
12
Views
283
  • Last Post
Replies
11
Views
507
  • Last Post
Replies
8
Views
1K
  • Last Post
Replies
1
Views
769
Replies
10
Views
848
Replies
29
Views
537
Replies
0
Views
2K
Replies
11
Views
2K
Replies
10
Views
990
Top