Relation between Mutual information and Expectation Values

[blueinfinity]

Homework Statement

Alice and Bob share the Bell state
\begin{align*}
|\psi\rangle = \frac{1}{\sqrt{2}}(|00\rangle+|11\rangle).
\end{align*}
Consider the pair of observables
\begin{align*}
\mathcal{O}_A =
\begin{pmatrix}
1 & 0 \\ 0 & \frac{1}{2}
\end{pmatrix}
, \qquad \mathcal{O}_B =
\begin{pmatrix}
1 & 0 \\ 0 & \frac{1}{3}
\end{pmatrix}
.
\end{align*}
Show the mutual information between Alice and Bob is larger than $(\langle\psi | \mathcal{O}_A \otimes \mathcal{O}_B| \psi\rangle - \langle\psi |\mathcal{O}_A|\psi \rangle \langle\psi |\mathcal{O}_B|\psi \rangle)^2 $

Relevant Equations

\begin{align*}
|\psi\rangle = \frac{1}{\sqrt{2}}(|00\rangle+|11\rangle).
\end{align*}
Consider the pair of observables
\begin{align*}
\mathcal{O}_A =
\begin{pmatrix}
1 & 0 \\ 0 & \frac{1}{2}
\end{pmatrix}
, \qquad \mathcal{O}_B =
\begin{pmatrix}
1 & 0 \\ 0 & \frac{1}{3}
\end{pmatrix}
.
\end{align*}

I've make progress in obtaining the values for the mutual information using the following:
$I(\rho_A:\rho_B) = S(\rho_A) +S(\rho_B) - S(\rho_{AB}) = 1 + 1 - 0 = 2.$

I would like to compute the expectation but I'm facing a problem in the case of $\langle\psi |\mathcal{O}_A|\psi \rangle$ since the size of matrices in this multiplication do not match. namely, $\langle\psi$ is of size $1\times 4$ and $|\psi \rangle$ is of size $4\times 1$ and the matrix $\mathcal{O}_A$ is $2 \times 2$.
I'm very new to the subject and I would greatly appreciate if I could have some guidance on how the computations for this expectation would be carried out.

additionally I have computed the $\langle\psi | \mathcal{O}_A \otimes \mathcal{O}_B| \psi\rangle$ by first computing the tensor product of the two matrices $A,B$ and then taken the multiplication with the Bra and Ket of the state respectively deducing
$$\langle\psi | \mathcal{O}_A \otimes \mathcal{O}_B| \psi\rangle = \frac{7}{12}$$.

I would appreciate any insight on this.

 

[PeroK]

You need two hashes as the delimiter for inline Latex.

 

[Hill]

Here it is edited:

Show the mutual information between Alice and Bob is larger than $(\langle\psi | \mathcal{O}_A \otimes \mathcal{O}_B| \psi\rangle - \langle\psi |\mathcal{O}_A|\psi \rangle \langle\psi |\mathcal{O}_B|\psi \rangle)^2$

I've make progress in obtaining the values for the mutual information using the following:
$I(\rho_A:\rho_B) = S(\rho_A) +S(\rho_B) - S(\rho_{AB}) = 1 + 1 - 0 = 2.$

I would like to compute the expectation but I'm facing a problem in the case of $\langle\psi |\mathcal{O}_A|\psi \rangle$ since the size of matrices in this multiplication do not match. namely, $\langle\psi$ is of size $1\times 4$ and $|\psi \rangle$ is of size $4\times 1$ and the matrix $\mathcal{O}_A$ is $2 \times 2$.
I'm very new to the subject and I would greatly appreciate if I could have some guidance on how the computations for this expectation would be carried out.

additionally I have computed the $\langle\psi | \mathcal{O}_A \otimes \mathcal{O}_B| \psi\rangle$ by first computing the tensor product of the two matrices $A,B$ and then taken the multiplication with the Bra and Ket of the state respectively deducing...

I think that for $\langle\psi |\mathcal{O}_A|\psi \rangle$ you calculate $\langle\psi | \mathcal{O}_A \otimes \mathcal I| \psi\rangle$, and for $\langle\psi |\mathcal{O}_B|\psi \rangle$, it is $\langle\psi |\mathcal I \otimes \mathcal{O}_B | \psi\rangle$.

(But I am new to it, too.)