Is \left\lfloor\Psi\right\rangle an Eigenstate of These Operators?

[doublemint]

Given:$$\left\lfloor\Psi\right\rangle$$=$$\frac{1}{\sqrt{2}}(\left{HHH}\right\rangle$$ + $$\left{VVV}\right\rangle)$$
show that it is an eigenstate of the following operators:

$$\hat{\sigma}_{x}\otimes\hat{\sigma}_{y}\otimes\hat{\sigma}_{y}$$
$$\hat{\sigma}_{y}\otimes\hat{\sigma}_{x}\otimes\hat{\sigma}_{y}$$
$$\hat{\sigma}_{y}\otimes\hat{\sigma}_{y}\otimes\hat{\sigma}_{x}$$
$$\hat{\sigma}_{x}\otimes\hat{\sigma}_{x}\otimes\hat{\sigma}_{x}$$
with eigenvalues, -1,-1,-1,1, respectively.

So what I did is completed the calculation for the tensors or the pauli matrices. Then I did the following: |Y><Y| to compare it to the operators. But |Y><Y|=(1/2)(2+|HHH><VVV|+|VVV><HHH|). So now I do not know what to do..
If anyone understands what I did and could provide some help, that would be much appreciated!
Thanks
DoubleMint

 

[grey_earl]

To be able to answer the question, you should provide some more information, like how your states $$|HHH\rangle$$ and $$|VVV\rangle$$ are defined. If not, I can only guess and that won't help you much ...

 

[dextercioby]

And also who's Y ? Please, post all the necessary details when asking for help, else people would first be asking you what you wanted to write about.

 

[doublemint]

The question is from this worksheet: http://qis.ucalgary.ca/quantech/443/2011/homework_four.pdf and is question 4.3c.
Grey_earl: The question only states that it is a Greenberg-Horne-Zeilinger state.
dextercioby: Sorry that i was not more detailed in my explanation, |Y> is |pis>, i just got lazy when posting.
So i solved, the tensor products of the pauli matrices but I don't know how to proceed from there.

 

[turab16]

hey double mint,

for the first question one from the same assignment, do we have to do in the matrices form or dirac notation?

 

[doublemint]

Hey turab16,
I did the first question in dirac notation. Are you in phys443 at ucalgary?

Oh and how far are you into this assignment?

 

[turab16]

yeah... and I suck at this stuff.. but i think for ur last question pm me ur email and i can send you something which might help you

 

[doublemint]

I know what you mean...I just pmed you, thanks for the help.

 

[turab16]

check ur email

 

[turab16]

how do u convert sigma y into H and V basis? for question 1 that is :S

 

[doublemint]

its in our notes and it should be $$\left|\hat{\sigma_{y}}$$=-i$$\left|{H}\right\rangle\langle\left{V}\right|$$ + i$$\left|{V}\right\rangle\langle\left{H}\right|$$

I still don't understand question 3 =( am i suppose to calculate the tensor product? unless there is an easy way to do it..

 

[doublemint]

Well me and turab16 could not get my originally posted question so if anyone could help us, that would be great!

 

[grey_earl]

You still haven't defined what those states are. Naming them "Greenberg-Horne-Zeilinger state" doesn't explain anything. The homework assignment obviously builds on your course notes where those states are defined, but to anyone else but you they are just meaningless names. The notation $$\sigma_x, \sigma_y, \sigma_z$$ for the Pauli matrices is standard, and you said in your original post that they are Pauli matrices, so everyone knows what they are, but your states $$|HHH\rangle$$ etc. is not.
So, if you want a reply, explain how they are defined exactly (look it up in your course notes!), and we may help you.

 

[doublemint]

Hey grey_earl,
The only explanation I have is that |H> and |V> are canonical basis so they are orthonormal to each other.

 

[sgd37]

and how do they act under the action of the pauli matricies are they eigenstates

 

[doublemint]

Well if we have three observers then the pauli matrices would represent the measurement of those three. So the tensor product says they are entangled. And since they are trying to measure the state Y ( sorry Y is state psi, I'm on my phone and i can't do fancy notation) the pauli matrices shoulds represent the eigenvalue and Y is the eigenstate?

 

[grey_earl]

$$|H\rangle$$ and $$|V\rangle$$ are the canonical basis of what? Canonical basis doesn't make sense without specifying the Hilbert space. But I think of all your posts I have enough information to give an answer that should be correct. Any mistakes go on your account of not fully defining the problem and variables.

So, the Hilbert space in question is the triple product space of a two-state system, for example electron spins. That is, each vector in the Hilbert space is a tensor product of three "spin" vectors which can take the states $$|H\rangle = (1, 0)$$ and $$|V\rangle = (0,1)$$. We then have $$\sigma_x |H\rangle = |V\rangle, \sigma_x |V\rangle = |H\rangle, \sigma_y |H\rangle = i |V\rangle, \sigma_y |V\rangle = - i |H\rangle, \sigma_z |H\rangle = |H\rangle, \sigma_z |V\rangle = - |V\rangle$$. Right? This should be in your course notes and this defines your states $$|H\rangle$$ and $$|V\rangle$$, and this you should have given in your first post.

Now just calculate:
$$\sigma_x \otimes \sigma_y \otimes \sigma_y |\psi\rangle = \frac{1}{\sqrt 2} \left[ \sigma_x \otimes \sigma_y \otimes \sigma_y |H\rangle \otimes |H\rangle \otimes |H\rangle + \sigma_x \otimes \sigma_y \otimes \sigma_y |V\rangle \otimes |V\rangle \otimes |V\rangle \right] = \frac{1}{\sqrt 2} \left[ - |V\rangle \otimes |V\rangle \otimes |V\rangle - |H\rangle \otimes |H\rangle \otimes |H\rangle \right] \right] = - |\psi\rangle$$,
so this state is an eigenstate of this operator with eigenvalue -1. The others are exactly the same.

 

[turab16]

I think you are right earl!. Thanks for your help!