How to find existence of quantum discord in two qubit state

[NotRealName]

Homework Statement

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.

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 don't understand how ψ_x 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

 

[mighty2000]

for any help!
Thank you for posting your question. It seems like you have a good understanding of the concept of tracing a subsystem and calculating eigenvalues and eigenvectors. However, I will try to provide a more detailed explanation of the process for your specific example.

First, let's define the state of your three-qubit system as ρ. In order to trace a subsystem, we need to take the partial trace over the qubits that we are not interested in, in this case qubit C. The resulting reduced density matrix for subsystem AB can be written as ρ_AB = Tr_C(ρ).

Now, let's calculate the eigenvalues and eigenvectors of ρ_AB. The eigenvalues of a density matrix represent the probabilities of obtaining a certain measurement outcome when measuring the state. In this case, we have four eigenvalues (3/4, 1/4, 0, 0) which correspond to the probabilities of obtaining the states |00>, |11>, |01>, and |10>, respectively.

Next, we need to find the eigenvectors of ρ_AB. These eigenvectors will be used to determine the discord in the system. The eigenvectors of ρ_AB can be found by solving the equation ρ_AB|ψ> = λ|ψ>, where λ is the corresponding eigenvalue and |ψ> is the eigenvector. In your example, the eigenvectors are (1,0,2,1), (-1,0,0,1), (1,0,-1,1), and (0,1,0,0) for the eigenvalues 3/4, 1/4, 0, and 0, respectively.

Now, we need to factorize these eigenvectors into two subsystems A and B. This is done by taking the first two elements of the eigenvector as the state of subsystem A and the last two elements as the state of subsystem B. For example, the first eigenvector (1,0,2,1) can be factorized into A = (1,0) and B = (2,1). Similarly, the other eigenvectors can be factorized into A and B.

Finally, we need to check if the matrices for subsystems A and B are pure, i.e. if the trace of their squared density matrices is equal to zero. If this