# How to find existence of quantum discord in two qubit state

1. May 18, 2013

### NotRealName

1. The problem statement, all variables and given/known data
Hello, I have a following problem. For a three-qubit state i need to trace subsystem. For this subsystem AB I calculate eigenvalues and eigenvectors. The task is now to determine according the eigenvalues and eigenvectors whether quantum discord in this system is non-zero

As far as i know, using this eigenvectors i should get a 2x2 matrix for each eigenvector and two subsystemsA,B. These matrices should be pure(if not discord is non-zero) such that Tr(ρ^2) = 0. Then I should write the matrix in a form |ρ>=α|0> + β|1> and second as well. These vectors should then have vector product which creates the original eigenvector. And this should be created for every eigenvector. However in this i might not be correct.

3. The attempt at a solution

I have an example however i still don't understand it. Having a GHZ state tracing the subsystem I get matrix 4x4 for AB defined as:
1/2 0 0 0
0 0 0 0
0 0 0 0
0 0 0 1/2
Now I have
|ψ1> = |00> for eigenvalue 1/2
|ψ2> = |11> for eigenvalue 1/2
|ψ3> = |01> for eigenvalue 0
|ψ4> = |10> for eigenvalue 0
these vectors are then factorized such that A has 0,1,0,1 and B 0,1,1,0 (I suppose that for A i take first part from vector and for B second)
Now I can see that
A=|0>,|1> they are orthogonal, which is same for B and the vectors for these systems are same,
thus discord is zero.

However I dont understand how ψx were found and if someone could explain it to me on a state:
0 0 0 0
0 1/3 1/3 0
0 1/3 1/3 0
0 0 0 1/3
where eigenvalues are 3/4,1/4,0,0 and eigenvectors are
(1,0,2,1),(-1,0,0,1),(1,0,-1,1),(0,1,0,0)
Thanks